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