let X be set ; :: thesis: ( X <> {} implies ex Y being set st
( Y in X & ( for 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 Y holds
Y1 misses X ) ) )
defpred S1[ 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 );
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 );
consider Z2 being set such that
A2: for Y being set holds
( Y in Z2 iff ( Y in union (union X) & S6[Y] ) ) from XBOOLE_0:sch 1();
 consider Z6 being set such that
A3: for Y being set holds
( Y in Z6 iff ( Y in union (union (union (union (union (union X))))) & S2[Y] ) ) from XBOOLE_0:sch 1();
 consider Z3 being set such that
A4: for Y being set holds
( Y in Z3 iff ( Y in union (union (union X)) & S5[Y] ) ) from XBOOLE_0:sch 1();
 consider Z5 being set such that
A5: for Y being set holds
( Y in Z5 iff ( Y in union (union (union (union (union X)))) & S3[Y] ) ) from XBOOLE_0:sch 1();
 consider Z4 being set such that
A6: for Y being set holds
( Y in Z4 iff ( Y in union (union (union (union X))) & S4[Y] ) ) from XBOOLE_0:sch 1();
 set V = (((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6;
assume X <> {} ; :: thesis: ex Y being set st
( Y in X & ( for 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 Y holds
Y1 misses X ) )
then consider Y being set such that
A7: Y in (((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6 and
A8: Y misses (((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6 by MCART_1:1;
A9: (((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6 = ((((X \/ (Z1 \/ Z2)) \/ Z3) \/ Z4) \/ Z5) \/ Z6 by XBOOLE_1:4
.= (((X \/ ((Z1 \/ Z2) \/ Z3)) \/ Z4) \/ Z5) \/ Z6 by XBOOLE_1:4
.= ((X \/ (((Z1 \/ Z2) \/ Z3) \/ Z4)) \/ Z5) \/ Z6 by XBOOLE_1:4
.= (X \/ ((((Z1 \/ Z2) \/ Z3) \/ Z4) \/ Z5)) \/ Z6 by XBOOLE_1:4
.= X \/ (((((Z1 \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6) by XBOOLE_1:4 ;
A10: now
assume A11: Y in Z1 ; :: thesis: contradiction
then consider Y1, Y2, Y3, Y4, Y5 being set such that
A12: ( Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 ) and
A13: Y5 in Y and
A14: Y1 meets X by A1;
Y in union X by A1, A11;
then Y5 in union (union X) by A13, TARSKI:def 4;
then Y5 in Z2 by A2, A12, A14;
then Y5 in (X \/ Z1) \/ Z2 by XBOOLE_0:def 3;
then Y meets (X \/ Z1) \/ Z2 by A13, 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;
hence contradiction by A8, XBOOLE_1:70; :: thesis: verum
end;
A15: now
assume A16: Y in Z2 ; :: thesis: contradiction
then consider Y1, Y2, Y3, Y4 being set such that
A17: ( Y1 in Y2 & Y2 in Y3 & Y3 in Y4 ) and
A18: Y4 in Y and
A19: Y1 meets X by A2;
Y in union (union X) by A2, A16;
then Y4 in union (union (union X)) by A18, TARSKI:def 4;
then Y4 in Z3 by A4, A17, A19;
then Y4 in ((X \/ Z1) \/ Z2) \/ Z3 by XBOOLE_0:def 3;
then Y4 in (((X \/ Z1) \/ Z2) \/ Z3) \/ Z4 by XBOOLE_0:def 3;
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;
hence contradiction by A8, A18, XBOOLE_0:3; :: thesis: verum
end;
A20: now
assume A21: Y in Z3 ; :: thesis: contradiction
then consider Y1, Y2, Y3 being set such that
A22: ( Y1 in Y2 & Y2 in Y3 ) and
A23: Y3 in Y and
A24: Y1 meets X by A4;
Y in union (union (union X)) by A4, A21;
then Y3 in union (union (union (union X))) by A23, TARSKI:def 4;
then Y3 in Z4 by A6, A22, A24;
then Y3 in (((X \/ Z1) \/ Z2) \/ Z3) \/ Z4 by XBOOLE_0:def 3;
then Y3 in ((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5 by XBOOLE_0:def 3;
then Y3 in (((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6 by XBOOLE_0:def 3;
hence contradiction by A8, A23, XBOOLE_0:3; :: thesis: verum
end;
A25: now
assume A26: Y in Z4 ; :: thesis: contradiction
then consider Y1, Y2 being set such that
A27: Y1 in Y2 and
A28: Y2 in Y and
A29: Y1 meets X by A6;
Y in union (union (union (union X))) by A6, A26;
then Y2 in union (union (union (union (union X)))) by A28, TARSKI:def 4;
then Y2 in Z5 by A5, A27, A29;
then Y2 in ((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5 by XBOOLE_0:def 3;
then Y2 in (((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6 by XBOOLE_0:def 3;
hence contradiction by A8, A28, XBOOLE_0:3; :: thesis: verum
end;
A30: now
assume A31: Y in Z5 ; :: thesis: contradiction
then consider Y1 being set such that
A32: Y1 in Y and
A33: Y1 meets X by A5;
Y in union (union (union (union (union X)))) by A5, A31;
then Y1 in union (union (union (union (union (union X))))) by A32, TARSKI:def 4;
then Y1 in Z6 by A3, A33;
then Y1 in (((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6 by XBOOLE_0:def 3;
hence contradiction by A8, A32, XBOOLE_0:3; :: thesis: verum
end;
assume A34: for Y being set holds
( not Y in X or 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 Y & not Y1 misses X ) ) ; :: thesis: contradiction
now
assume A35: Y in X ; :: thesis: contradiction
then consider Y1, Y2, Y3, Y4, Y5, Y6 being set such that
A36: ( Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 ) and
A37: Y6 in Y and
A38: not Y1 misses X by A34;
Y6 in union X by A35, A37, TARSKI:def 4;
then Y6 in Z1 by A1, A36, A38;
then Y6 in X \/ Z1 by XBOOLE_0:def 3;
then Y6 in (X \/ Z1) \/ Z2 by XBOOLE_0:def 3;
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 Y meets (((X \/ Z1) \/ Z2) \/ Z3) \/ Z4 by A37, XBOOLE_0:3;
then Y meets ((((X \/ Z1) \/ Z2) \/ Z3) \/ Z4) \/ Z5 by XBOOLE_1:70;
hence contradiction by A8, XBOOLE_1:70; :: thesis: verum
end;
then Y in ((((Z1 \/ Z2) \/ Z3) \/ Z4) \/ Z5) \/ Z6 by A9, A7, XBOOLE_0:def 3;
then Y in (((Z1 \/ (Z2 \/ Z3)) \/ Z4) \/ Z5) \/ Z6 by XBOOLE_1:4;
then Y in ((Z1 \/ ((Z2 \/ Z3) \/ Z4)) \/ Z5) \/ Z6 by XBOOLE_1:4;
then Y in (Z1 \/ (((Z2 \/ Z3) \/ Z4) \/ Z5)) \/ Z6 by XBOOLE_1:4;
then Y in Z1 \/ ((((Z2 \/ Z3) \/ Z4) \/ Z5) \/ Z6) by XBOOLE_1:4;
then Y in (((Z2 \/ Z3) \/ Z4) \/ Z5) \/ Z6 by A10, XBOOLE_0:def 3;
then Y in ((Z2 \/ (Z3 \/ Z4)) \/ Z5) \/ Z6 by XBOOLE_1:4;
then Y in (Z2 \/ ((Z3 \/ Z4) \/ Z5)) \/ Z6 by XBOOLE_1:4;
then Y in Z2 \/ (((Z3 \/ Z4) \/ Z5) \/ Z6) by XBOOLE_1:4;
then Y in ((Z3 \/ Z4) \/ Z5) \/ Z6 by A15, XBOOLE_0:def 3;
then Y in (Z3 \/ (Z4 \/ Z5)) \/ Z6 by XBOOLE_1:4;
then Y in Z3 \/ ((Z4 \/ Z5) \/ Z6) by XBOOLE_1:4;
then Y in (Z4 \/ Z5) \/ Z6 by A20, XBOOLE_0:def 3;
then Y in Z4 \/ (Z5 \/ Z6) by XBOOLE_1:4;
then Y in Z5 \/ Z6 by A25, XBOOLE_0:def 3;
then Y in Z6 by A30, XBOOLE_0:def 3;
then Y meets X by A3;
hence contradiction by A9, A8, XBOOLE_1:70; :: thesis: verum