let X be set ; :: thesis: ( X <> {} implies ex Y being set st
( Y in X & ( for Y1, Y2, Y3, Y4, Y5, Y6, Y7, Y8, Y9, YA, YB, YC, YD being set st Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7 & Y7 in Y8 & Y8 in Y9 & Y9 in YA & YA in YB & YB in YC & YC in YD & YD in Y holds
Y1 misses X ) ) )
defpred S1[ set ] means ex Y1, Y2, Y3, Y4, Y5, Y6, Y7, Y8, Y9, YA, YB, YC being set st
( Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7 & Y7 in Y8 & Y8 in Y9 & Y9 in YA & YA in YB & YB in YC & YC in $1 & Y1 meets X );
consider Z1 being set such that
A1: for Y being set holds
( Y in Z1 iff ( Y in union X & S1[Y] ) ) from XBOOLE_0:sch 1();
 defpred S2[ set ] means $1 meets X;
defpred S3[ set ] means ex Y1 being set st
( Y1 in $1 & Y1 meets X );
defpred S4[ set ] means ex Y1, Y2 being set st
( Y1 in Y2 & Y2 in $1 & Y1 meets X );
defpred S5[ set ] means ex Y1, Y2, Y3 being set st
( Y1 in Y2 & Y2 in Y3 & Y3 in $1 & Y1 meets X );
defpred S6[ set ] means ex Y1, Y2, Y3, Y4 being set st
( Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in $1 & Y1 meets X );
defpred S7[ set ] means ex Y1, Y2, Y3, Y4, Y5 being set st
( Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in $1 & Y1 meets X );
defpred S8[ set ] means ex Y1, Y2, Y3, Y4, Y5, Y6 being set st
( Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in $1 & Y1 meets X );
defpred S9[ set ] means ex Y1, Y2, Y3, Y4, Y5, Y6, Y7 being set st
( Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7 & Y7 in $1 & Y1 meets X );
defpred S10[ set ] means ex Y1, Y2, Y3, Y4, Y5, Y6, Y7, Y8 being set st
( Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7 & Y7 in Y8 & Y8 in $1 & Y1 meets X );
defpred S11[ set ] means ex Y1, Y2, Y3, Y4, Y5, Y6, Y7, Y8, Y9 being set st
( Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7 & Y7 in Y8 & Y8 in Y9 & Y9 in $1 & Y1 meets X );
defpred S12[ set ] means ex Y1, Y2, Y3, Y4, Y5, Y6, Y7, Y8, Y9, YA being set st
( Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7 & Y7 in Y8 & Y8 in Y9 & Y9 in YA & YA in $1 & Y1 meets X );
defpred S13[ set ] means ex Y1, Y2, Y3, Y4, Y5, Y6, Y7, Y8, Y9, YA, YB being set st
( Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7 & Y7 in Y8 & Y8 in Y9 & Y9 in YA & YA in YB & YB in $1 & Y1 meets X );
consider Z2 being set such that
A2: for Y being set holds
( Y in Z2 iff ( Y in union (union X) & S13[Y] ) ) from XBOOLE_0:sch 1();
 consider Z9 being set such that
A3: for Y being set holds
( Y in Z9 iff ( Y in union (union (union (union (union (union (union (union (union X)))))))) & S6[Y] ) ) from XBOOLE_0:sch 1();
 consider Z8 being set such that
A4: for Y being set holds
( Y in Z8 iff ( Y in union (union (union (union (union (union (union (union X))))))) & S7[Y] ) ) from XBOOLE_0:sch 1();
 consider ZD being set such that
A5: for Y being set holds
( Y in ZD iff ( Y in union (union (union (union (union (union (union (union (union (union (union (union (union X)))))))))))) & S2[Y] ) ) from XBOOLE_0:sch 1();
 consider Z3 being set such that
A6: for Y being set holds
( Y in Z3 iff ( Y in union (union (union X)) & S12[Y] ) ) from XBOOLE_0:sch 1();
 consider Z7 being set such that
A7: for Y being set holds
( Y in Z7 iff ( Y in union (union (union (union (union (union (union X)))))) & S8[Y] ) ) from XBOOLE_0:sch 1();
 consider Z6 being set such that
A8: for Y being set holds
( Y in Z6 iff ( Y in union (union (union (union (union (union X))))) & S9[Y] ) ) from XBOOLE_0:sch 1();
 consider ZB being set such that
A9: for Y being set holds
( Y in ZB iff ( Y in union (union (union (union (union (union (union (union (union (union (union X)))))))))) & S4[Y] ) ) from XBOOLE_0:sch 1();
 consider ZA being set such that
A10: for Y being set holds
( Y in ZA iff ( Y in union (union (union (union (union (union (union (union (union (union X))))))))) & S5[Y] ) ) from XBOOLE_0:sch 1();
 consider ZC being set such that
A11: for Y being set holds
( Y in ZC iff ( Y in union (union (union (union (union (union (union (union (union (union (union (union X))))))))))) & S3[Y] ) ) from XBOOLE_0:sch 1();
 consider Z5 being set such that
A12: for Y being set holds
( Y in Z5 iff ( Y in union (union (union (union (union X)))) & S10[Y] ) ) from XBOOLE_0:sch 1();
 consider Z4 being set such that
A13: for Y being set holds
( Y in Z4 iff ( Y in union (union (union (union X))) & S11[Y] ) ) from XBOOLE_0:sch 1();
 set V = ((((((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA) \/ ZB) \/ ZC) \/ ZD;
A14: ((((((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA) \/ ZB) \/ ZC) \/ ZD = (((((((((((X \/ (Z1 \/ Z2)) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA) \/ ZB) \/ ZC) \/ ZD by XBOOLE_1:4
.= ((((((((((X \/ ((Z1 \/ Z2) \/ Z3)) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA) \/ ZB) \/ ZC) \/ ZD by XBOOLE_1:4
.= (((((((((X \/ (((Z1 \/ Z2) \/ Z3) \/ Z4)) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA) \/ ZB) \/ ZC) \/ ZD by XBOOLE_1:4
.= ((((((((X \/ ((((Z1 \/ Z2) \/ Z3) \/ Z4) \/ Z5)) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA) \/ ZB) \/ ZC) \/ ZD by XBOOLE_1:4
.= (((((((X \/ (((((Z1 \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6)) \/ Z7) \/ Z8) \/ Z9) \/ ZA) \/ ZB) \/ ZC) \/ ZD by XBOOLE_1:4
.= ((((((X \/ ((((((Z1 \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7)) \/ Z8) \/ Z9) \/ ZA) \/ ZB) \/ ZC) \/ ZD by XBOOLE_1:4
.= (((((X \/ (((((((Z1 \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8)) \/ Z9) \/ ZA) \/ ZB) \/ ZC) \/ ZD by XBOOLE_1:4
.= ((((X \/ ((((((((Z1 \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9)) \/ ZA) \/ ZB) \/ ZC) \/ ZD by XBOOLE_1:4
.= (((X \/ (((((((((Z1 \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA)) \/ ZB) \/ ZC) \/ ZD by XBOOLE_1:4
.= ((X \/ ((((((((((Z1 \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA) \/ ZB)) \/ ZC) \/ ZD by XBOOLE_1:4
.= (X \/ (((((((((((Z1 \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA) \/ ZB) \/ ZC)) \/ ZD by XBOOLE_1:4
.= X \/ ((((((((((((Z1 \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA) \/ ZB) \/ ZC) \/ ZD) by XBOOLE_1:4 ;
assume X <> {} ; :: thesis: ex Y being set st
( Y in X & ( for Y1, Y2, Y3, Y4, Y5, Y6, Y7, Y8, Y9, YA, YB, YC, YD being set st Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7 & Y7 in Y8 & Y8 in Y9 & Y9 in YA & YA in YB & YB in YC & YC in YD & YD in Y holds
Y1 misses X ) )
then consider Y being set such that
A15: Y in ((((((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA) \/ ZB) \/ ZC) \/ ZD and
A16: Y misses ((((((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA) \/ ZB) \/ ZC) \/ ZD by MCART_1:1;
( Y in (((((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA) \/ ZB) \/ ZC or Y in ZD ) by A15, XBOOLE_0:def 3;
then ( Y in ((((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA) \/ ZB or Y in ZC or Y in ZD ) by XBOOLE_0:def 3;
then ( Y in (((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA or Y in ZB or Y in ZC or Y in ZD ) by XBOOLE_0:def 3;
then ( Y in ((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9 or Y in ZA or Y in ZB or Y in ZC or Y in ZD ) by XBOOLE_0:def 3;
then ( Y in (((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8 or Y in Z9 or Y in ZA or Y in ZB or Y in ZC or Y in ZD ) by XBOOLE_0:def 3;
then ( Y in ((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7 or Y in Z8 or Y in Z9 or Y in ZA or Y in ZB or Y in ZC or Y in ZD ) by XBOOLE_0:def 3;
then ( Y in (((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6 or Y in Z7 or Y in Z8 or Y in Z9 or Y in ZA or Y in ZB or Y in ZC or Y in ZD ) by XBOOLE_0:def 3;
then ( Y in ((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5 or Y in Z6 or Y in Z7 or Y in Z8 or Y in Z9 or Y in ZA or Y in ZB or Y in ZC or Y in ZD ) by XBOOLE_0:def 3;
then ( Y in (((X \/ Z1) \/ Z2) \/ Z3) \/ Z4 or Y in Z5 or Y in Z6 or Y in Z7 or Y in Z8 or Y in Z9 or Y in ZA or Y in ZB or Y in ZC or Y in ZD ) by XBOOLE_0:def 3;
then ( Y in ((X \/ Z1) \/ Z2) \/ Z3 or Y in Z4 or Y in Z5 or Y in Z6 or Y in Z7 or Y in Z8 or Y in Z9 or Y in ZA or Y in ZB or Y in ZC or Y in ZD ) by XBOOLE_0:def 3;
then ( Y in (X \/ Z1) \/ Z2 or Y in Z3 or Y in Z4 or Y in Z5 or Y in Z6 or Y in Z7 or Y in Z8 or Y in Z9 or Y in ZA or Y in ZB or Y in ZC or Y in ZD ) by XBOOLE_0:def 3;
then A17: ( Y in X \/ Z1 or Y in Z2 or Y in Z3 or Y in Z4 or Y in Z5 or Y in Z6 or Y in Z7 or Y in Z8 or Y in Z9 or Y in ZA or Y in ZB or Y in ZC or Y in ZD ) by XBOOLE_0:def 3;
assume A18: for Y being set holds
( not Y in X or ex Y1, Y2, Y3, Y4, Y5, Y6, Y7, Y8, Y9, YA, YB, YC, YD being set st
( Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7 & Y7 in Y8 & Y8 in Y9 & Y9 in YA & YA in YB & YB in YC & YC in YD & YD in Y & not Y1 misses X ) ) ; :: thesis: contradiction
per cases ( Y in X or Y in Z1 or Y in Z2 or Y in Z3 or Y in Z4 or Y in Z5 or Y in Z6 or Y in Z7 or Y in Z8 or Y in Z9 or Y in ZA or Y in ZB or Y in ZC or Y in ZD ) by A17, XBOOLE_0:def 3;
suppose A19: Y in X ; :: thesis: contradiction
then consider Y1, Y2, Y3, Y4, Y5, Y6, Y7, Y8, Y9, YA, YB, YC, YD being set such that
A20: ( Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7 & Y7 in Y8 & Y8 in Y9 & Y9 in YA & YA in YB & YB in YC & YC in YD ) and
A21: YD in Y and
A22: not Y1 misses X by A18;
YD in union X by A19, A21, TARSKI:def 4;
then YD in Z1 by A1, A20, A22;
then YD in X \/ Z1 by XBOOLE_0:def 3;
then YD in (X \/ Z1) \/ Z2 by XBOOLE_0:def 3;
then YD in ((X \/ Z1) \/ Z2) \/ Z3 by XBOOLE_0:def 3;
then YD in (((X \/ Z1) \/ Z2) \/ Z3) \/ Z4 by XBOOLE_0:def 3;
then YD in ((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5 by XBOOLE_0:def 3;
then Y meets ((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5 by A21, XBOOLE_0:3;
then Y meets (((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6 by XBOOLE_1:70;
then Y meets ((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7 by XBOOLE_1:70;
then Y meets (((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8 by XBOOLE_1:70;
then Y meets ((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9 by XBOOLE_1:70;
then Y meets (((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA by XBOOLE_1:70;
then Y meets ((((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA) \/ ZB by XBOOLE_1:70;
then Y meets (((((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA) \/ ZB) \/ ZC by XBOOLE_1:70;
hence contradiction by A16, XBOOLE_1:70; :: thesis: verum
end;
suppose A23: Y in Z1 ; :: thesis: contradiction
then consider Y1, Y2, Y3, Y4, Y5, Y6, Y7, Y8, Y9, YA, YB, YC being set such that
A24: ( Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7 & Y7 in Y8 & Y8 in Y9 & Y9 in YA & YA in YB & YB in YC ) and
A25: YC in Y and
A26: Y1 meets X by A1;
Y in union X by A1, A23;
then YC in union (union X) by A25, TARSKI:def 4;
then YC in Z2 by A2, A24, A26;
then YC in (X \/ Z1) \/ Z2 by XBOOLE_0:def 3;
then Y meets (X \/ Z1) \/ Z2 by A25, XBOOLE_0:3;
then Y meets ((X \/ Z1) \/ Z2) \/ Z3 by XBOOLE_1:70;
then Y meets (((X \/ Z1) \/ Z2) \/ Z3) \/ Z4 by XBOOLE_1:70;
then Y meets ((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5 by XBOOLE_1:70;
then Y meets (((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6 by XBOOLE_1:70;
then Y meets ((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7 by XBOOLE_1:70;
then Y meets (((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8 by XBOOLE_1:70;
then Y meets ((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9 by XBOOLE_1:70;
then Y meets (((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA by XBOOLE_1:70;
then Y meets ((((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA) \/ ZB by XBOOLE_1:70;
then Y meets (((((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA) \/ ZB) \/ ZC by XBOOLE_1:70;
hence contradiction by A16, XBOOLE_1:70; :: thesis: verum
end;
suppose A27: Y in Z2 ; :: thesis: contradiction
then consider Y1, Y2, Y3, Y4, Y5, Y6, Y7, Y8, Y9, YA, YB being set such that
A28: ( Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7 & Y7 in Y8 & Y8 in Y9 & Y9 in YA & YA in YB ) and
A29: YB in Y and
A30: Y1 meets X by A2;
Y in union (union X) by A2, A27;
then YB in union (union (union X)) by A29, TARSKI:def 4;
then YB in Z3 by A6, A28, A30;
then YB in ((X \/ Z1) \/ Z2) \/ Z3 by XBOOLE_0:def 3;
then YB in (((X \/ Z1) \/ Z2) \/ Z3) \/ Z4 by XBOOLE_0:def 3;
then YB in ((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5 by XBOOLE_0:def 3;
then YB in (((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6 by XBOOLE_0:def 3;
then YB in ((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7 by XBOOLE_0:def 3;
then YB in (((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8 by XBOOLE_0:def 3;
then YB in ((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9 by XBOOLE_0:def 3;
then YB in (((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA by XBOOLE_0:def 3;
then YB in ((((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA) \/ ZB by XBOOLE_0:def 3;
then YB in (((((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA) \/ ZB) \/ ZC by XBOOLE_0:def 3;
then YB in ((((((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA) \/ ZB) \/ ZC) \/ ZD by XBOOLE_0:def 3;
hence contradiction by A16, A29, XBOOLE_0:3; :: thesis: verum
end;
suppose A31: Y in Z3 ; :: thesis: contradiction
then consider Y1, Y2, Y3, Y4, Y5, Y6, Y7, Y8, Y9, YA being set such that
A32: ( Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7 & Y7 in Y8 & Y8 in Y9 & Y9 in YA ) and
A33: YA in Y and
A34: Y1 meets X by A6;
Y in union (union (union X)) by A6, A31;
then YA in union (union (union (union X))) by A33, TARSKI:def 4;
then YA in Z4 by A13, A32, A34;
then YA in (((X \/ Z1) \/ Z2) \/ Z3) \/ Z4 by XBOOLE_0:def 3;
then YA in ((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5 by XBOOLE_0:def 3;
then YA in (((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6 by XBOOLE_0:def 3;
then YA in ((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7 by XBOOLE_0:def 3;
then YA in (((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8 by XBOOLE_0:def 3;
then YA in ((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9 by XBOOLE_0:def 3;
then YA in (((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA by XBOOLE_0:def 3;
then YA in ((((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA) \/ ZB by XBOOLE_0:def 3;
then YA in (((((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA) \/ ZB) \/ ZC by XBOOLE_0:def 3;
then YA in ((((((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA) \/ ZB) \/ ZC) \/ ZD by XBOOLE_0:def 3;
hence contradiction by A16, A33, XBOOLE_0:3; :: thesis: verum
end;
suppose A35: Y in Z4 ; :: thesis: contradiction
then consider Y1, Y2, Y3, Y4, Y5, Y6, Y7, Y8, Y9 being set such that
A36: ( Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7 & Y7 in Y8 & Y8 in Y9 ) and
A37: Y9 in Y and
A38: Y1 meets X by A13;
Y in union (union (union (union X))) by A13, A35;
then Y9 in union (union (union (union (union X)))) by A37, TARSKI:def 4;
then Y9 in Z5 by A12, A36, A38;
then Y9 in ((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5 by XBOOLE_0:def 3;
then Y9 in (((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6 by XBOOLE_0:def 3;
then Y9 in ((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7 by XBOOLE_0:def 3;
then Y9 in (((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8 by XBOOLE_0:def 3;
then Y9 in ((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9 by XBOOLE_0:def 3;
then Y9 in (((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA by XBOOLE_0:def 3;
then Y9 in ((((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA) \/ ZB by XBOOLE_0:def 3;
then Y9 in (((((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA) \/ ZB) \/ ZC by XBOOLE_0:def 3;
then Y9 in ((((((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA) \/ ZB) \/ ZC) \/ ZD by XBOOLE_0:def 3;
hence contradiction by A16, A37, XBOOLE_0:3; :: thesis: verum
end;
suppose A39: Y in Z5 ; :: thesis: contradiction
then consider Y1, Y2, Y3, Y4, Y5, Y6, Y7, Y8 being set such that
A40: ( Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7 & Y7 in Y8 ) and
A41: Y8 in Y and
A42: Y1 meets X by A12;
Y in union (union (union (union (union X)))) by A12, A39;
then Y8 in union (union (union (union (union (union X))))) by A41, TARSKI:def 4;
then Y8 in Z6 by A8, A40, A42;
then Y8 in (((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6 by XBOOLE_0:def 3;
then Y8 in ((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7 by XBOOLE_0:def 3;
then Y8 in (((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8 by XBOOLE_0:def 3;
then Y8 in ((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9 by XBOOLE_0:def 3;
then Y8 in (((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA by XBOOLE_0:def 3;
then Y8 in ((((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA) \/ ZB by XBOOLE_0:def 3;
then Y8 in (((((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA) \/ ZB) \/ ZC by XBOOLE_0:def 3;
then Y8 in ((((((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA) \/ ZB) \/ ZC) \/ ZD by XBOOLE_0:def 3;
hence contradiction by A16, A41, XBOOLE_0:3; :: thesis: verum
end;
suppose A43: Y in Z6 ; :: thesis: contradiction
then consider Y1, Y2, Y3, Y4, Y5, Y6, Y7 being set such that
A44: ( Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7 ) and
A45: Y7 in Y and
A46: Y1 meets X by A8;
Y in union (union (union (union (union (union X))))) by A8, A43;
then Y7 in union (union (union (union (union (union (union X)))))) by A45, TARSKI:def 4;
then Y7 in Z7 by A7, A44, A46;
then Y7 in ((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7 by XBOOLE_0:def 3;
then Y7 in (((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8 by XBOOLE_0:def 3;
then Y7 in ((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9 by XBOOLE_0:def 3;
then Y7 in (((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA by XBOOLE_0:def 3;
then Y7 in ((((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA) \/ ZB by XBOOLE_0:def 3;
then Y7 in (((((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA) \/ ZB) \/ ZC by XBOOLE_0:def 3;
then Y7 in ((((((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA) \/ ZB) \/ ZC) \/ ZD by XBOOLE_0:def 3;
hence contradiction by A16, A45, XBOOLE_0:3; :: thesis: verum
end;
suppose A47: Y in Z7 ; :: thesis: contradiction
then consider Y1, Y2, Y3, Y4, Y5, Y6 being set such that
A48: ( Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 ) and
A49: Y6 in Y and
A50: Y1 meets X by A7;
Y in union (union (union (union (union (union (union X)))))) by A7, A47;
then Y6 in union (union (union (union (union (union (union (union X))))))) by A49, TARSKI:def 4;
then Y6 in Z8 by A4, A48, A50;
then Y6 in (((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8 by XBOOLE_0:def 3;
then Y6 in ((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9 by XBOOLE_0:def 3;
then Y6 in (((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA by XBOOLE_0:def 3;
then Y6 in ((((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA) \/ ZB by XBOOLE_0:def 3;
then Y6 in (((((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA) \/ ZB) \/ ZC by XBOOLE_0:def 3;
then Y6 in ((((((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA) \/ ZB) \/ ZC) \/ ZD by XBOOLE_0:def 3;
hence contradiction by A16, A49, XBOOLE_0:3; :: thesis: verum
end;
suppose A51: Y in Z8 ; :: thesis: contradiction
then consider Y1, Y2, Y3, Y4, Y5 being set such that
A52: ( Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 ) and
A53: Y5 in Y and
A54: Y1 meets X by A4;
Y in union (union (union (union (union (union (union (union X))))))) by A4, A51;
then Y5 in union (union (union (union (union (union (union (union (union X)))))))) by A53, TARSKI:def 4;
then Y5 in Z9 by A3, A52, A54;
then Y5 in ((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9 by XBOOLE_0:def 3;
then Y5 in (((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA by XBOOLE_0:def 3;
then Y5 in ((((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA) \/ ZB by XBOOLE_0:def 3;
then Y5 in (((((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA) \/ ZB) \/ ZC by XBOOLE_0:def 3;
then Y5 in ((((((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA) \/ ZB) \/ ZC) \/ ZD by XBOOLE_0:def 3;
hence contradiction by A16, A53, XBOOLE_0:3; :: thesis: verum
end;
suppose A55: Y in Z9 ; :: thesis: contradiction
then consider Y1, Y2, Y3, Y4 being set such that
A56: ( Y1 in Y2 & Y2 in Y3 & Y3 in Y4 ) and
A57: Y4 in Y and
A58: Y1 meets X by A3;
Y in union (union (union (union (union (union (union (union (union X)))))))) by A3, A55;
then Y4 in union (union (union (union (union (union (union (union (union (union X))))))))) by A57, TARSKI:def 4;
then Y4 in ZA by A10, A56, A58;
then Y4 in (((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA by XBOOLE_0:def 3;
then Y4 in ((((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA) \/ ZB by XBOOLE_0:def 3;
then Y4 in (((((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA) \/ ZB) \/ ZC by XBOOLE_0:def 3;
then Y4 in ((((((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA) \/ ZB) \/ ZC) \/ ZD by XBOOLE_0:def 3;
hence contradiction by A16, A57, XBOOLE_0:3; :: thesis: verum
end;
suppose A59: Y in ZA ; :: thesis: contradiction
then consider Y1, Y2, Y3 being set such that
A60: ( Y1 in Y2 & Y2 in Y3 ) and
A61: Y3 in Y and
A62: Y1 meets X by A10;
Y in union (union (union (union (union (union (union (union (union (union X))))))))) by A10, A59;
then Y3 in union (union (union (union (union (union (union (union (union (union (union X)))))))))) by A61, TARSKI:def 4;
then Y3 in ZB by A9, A60, A62;
then Y3 in ((((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA) \/ ZB by XBOOLE_0:def 3;
then Y3 in (((((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA) \/ ZB) \/ ZC by XBOOLE_0:def 3;
then Y3 in ((((((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA) \/ ZB) \/ ZC) \/ ZD by XBOOLE_0:def 3;
hence contradiction by A16, A61, XBOOLE_0:3; :: thesis: verum
end;
suppose A63: Y in ZB ; :: thesis: contradiction
then consider Y1, Y2 being set such that
A64: Y1 in Y2 and
A65: Y2 in Y and
A66: Y1 meets X by A9;
Y in union (union (union (union (union (union (union (union (union (union (union X)))))))))) by A9, A63;
then Y2 in union (union (union (union (union (union (union (union (union (union (union (union X))))))))))) by A65, TARSKI:def 4;
then Y2 in ZC by A11, A64, A66;
then Y2 in (((((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA) \/ ZB) \/ ZC by XBOOLE_0:def 3;
then Y2 in ((((((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA) \/ ZB) \/ ZC) \/ ZD by XBOOLE_0:def 3;
hence contradiction by A16, A65, XBOOLE_0:3; :: thesis: verum
end;
suppose A67: Y in ZC ; :: thesis: contradiction
then consider Y1 being set such that
A68: Y1 in Y and
A69: Y1 meets X by A11;
Y in union (union (union (union (union (union (union (union (union (union (union (union X))))))))))) by A11, A67;
then Y1 in union (union (union (union (union (union (union (union (union (union (union (union (union X)))))))))))) by A68, TARSKI:def 4;
then Y1 in ZD by A5, A69;
then Y1 in ((((((((((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) \/ Z7) \/ Z8) \/ Z9) \/ ZA) \/ ZB) \/ ZC) \/ ZD by XBOOLE_0:def 3;
hence contradiction by A16, A68, XBOOLE_0:3; :: thesis: verum
end;
suppose Y in ZD ; :: thesis: contradiction
then Y meets X by A5;
hence contradiction by A14, A16, XBOOLE_1:70; :: thesis: verum
end;
end;