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