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