let X be set ; :: thesis: ( X <> {} implies ex v being set st
( v in X & ( for x1, x2, x3, x4, x5, x6 being set holds
( ( not x1 in X & not x2 in X ) or not v = [x1,x2,x3,x4,x5,x6] ) ) ) )
assume X <> {} ; :: thesis: ex v being set st
( v in X & ( for x1, x2, x3, x4, x5, x6 being set holds
( ( not x1 in X & not x2 in X ) or not v = [x1,x2,x3,x4,x5,x6] ) ) )
then consider Y being set such that
A1: Y in X and
A2: for 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 Y holds
Y1 misses X by Th40;
take v = Y; :: thesis: ( v in X & ( for x1, x2, x3, x4, x5, x6 being set holds
( ( not x1 in X & not x2 in X ) or not v = [x1,x2,x3,x4,x5,x6] ) ) )
thus v in X by A1; :: thesis: for x1, x2, x3, x4, x5, x6 being set holds
( ( not x1 in X & not x2 in X ) or not v = [x1,x2,x3,x4,x5,x6] )
given x1, x2, x3, x4, x5, x6 being set such that A3: ( x1 in X or x2 in X ) and
A4: v = [x1,x2,x3,x4,x5,x6] ; :: thesis: contradiction
set Y1 = {x1,x2};
set Y2 = {{x1,x2},{x1}};
set Y3 = {{{x1,x2},{x1}},x3};
set Y4 = {{{{x1,x2},{x1}},x3},{{{x1,x2},{x1}}}};
set Y5 = {{{{{x1,x2},{x1}},x3},{{{x1,x2},{x1}}}},x4};
set Y6 = {{{{{{x1,x2},{x1}},x3},{{{x1,x2},{x1}}}},x4},{{{{{x1,x2},{x1}},x3},{{{x1,x2},{x1}}}}}};
set Y7 = {{{{{{{x1,x2},{x1}},x3},{{{x1,x2},{x1}}}},x4},{{{{{x1,x2},{x1}},x3},{{{x1,x2},{x1}}}}}},x5};
set Y8 = {{{{{{{{x1,x2},{x1}},x3},{{{x1,x2},{x1}}}},x4},{{{{{x1,x2},{x1}},x3},{{{x1,x2},{x1}}}}}},x5},{{{{{{{x1,x2},{x1}},x3},{{{x1,x2},{x1}}}},x4},{{{{{x1,x2},{x1}},x3},{{{x1,x2},{x1}}}}}}}};
set Y9 = {{{{{{{{{x1,x2},{x1}},x3},{{{x1,x2},{x1}}}},x4},{{{{{x1,x2},{x1}},x3},{{{x1,x2},{x1}}}}}},x5},{{{{{{{x1,x2},{x1}},x3},{{{x1,x2},{x1}}}},x4},{{{{{x1,x2},{x1}},x3},{{{x1,x2},{x1}}}}}}}},x6};
A5: ( {{{x1,x2},{x1}},x3} in {{{{x1,x2},{x1}},x3},{{{x1,x2},{x1}}}} & {{{{x1,x2},{x1}},x3},{{{x1,x2},{x1}}}} in {{{{{x1,x2},{x1}},x3},{{{x1,x2},{x1}}}},x4} ) by TARSKI:def 2;
( x1 in {x1,x2} & x2 in {x1,x2} ) by TARSKI:def 2;
then A6: not {x1,x2} misses X by A3, XBOOLE_0:3;
A7: ( {{{{{{{x1,x2},{x1}},x3},{{{x1,x2},{x1}}}},x4},{{{{{x1,x2},{x1}},x3},{{{x1,x2},{x1}}}}}},x5} in {{{{{{{{x1,x2},{x1}},x3},{{{x1,x2},{x1}}}},x4},{{{{{x1,x2},{x1}},x3},{{{x1,x2},{x1}}}}}},x5},{{{{{{{x1,x2},{x1}},x3},{{{x1,x2},{x1}}}},x4},{{{{{x1,x2},{x1}},x3},{{{x1,x2},{x1}}}}}}}} & {{{{{{{{x1,x2},{x1}},x3},{{{x1,x2},{x1}}}},x4},{{{{{x1,x2},{x1}},x3},{{{x1,x2},{x1}}}}}},x5},{{{{{{{x1,x2},{x1}},x3},{{{x1,x2},{x1}}}},x4},{{{{{x1,x2},{x1}},x3},{{{x1,x2},{x1}}}}}}}} in {{{{{{{{{x1,x2},{x1}},x3},{{{x1,x2},{x1}}}},x4},{{{{{x1,x2},{x1}},x3},{{{x1,x2},{x1}}}}}},x5},{{{{{{{x1,x2},{x1}},x3},{{{x1,x2},{x1}}}},x4},{{{{{x1,x2},{x1}},x3},{{{x1,x2},{x1}}}}}}}},x6} ) by TARSKI:def 2;
Y = [[[[[x1,x2],x3],x4],x5],x6] by A4, Th41
.= [[[[{{x1,x2},{x1}},x3],x4],x5],x6] by TARSKI:def 5
.= [[[{{{{x1,x2},{x1}},x3},{{{x1,x2},{x1}}}},x4],x5],x6] by TARSKI:def 5
.= [[{{{{{{x1,x2},{x1}},x3},{{{x1,x2},{x1}}}},x4},{{{{{x1,x2},{x1}},x3},{{{x1,x2},{x1}}}}}},x5],x6] by TARSKI:def 5
.= [{{{{{{{{x1,x2},{x1}},x3},{{{x1,x2},{x1}}}},x4},{{{{{x1,x2},{x1}},x3},{{{x1,x2},{x1}}}}}},x5},{{{{{{{x1,x2},{x1}},x3},{{{x1,x2},{x1}}}},x4},{{{{{x1,x2},{x1}},x3},{{{x1,x2},{x1}}}}}}}},x6] by TARSKI:def 5
.= {{{{{{{{{{x1,x2},{x1}},x3},{{{x1,x2},{x1}}}},x4},{{{{{x1,x2},{x1}},x3},{{{x1,x2},{x1}}}}}},x5},{{{{{{{x1,x2},{x1}},x3},{{{x1,x2},{x1}}}},x4},{{{{{x1,x2},{x1}},x3},{{{x1,x2},{x1}}}}}}}},x6},{{{{{{{{{x1,x2},{x1}},x3},{{{x1,x2},{x1}}}},x4},{{{{{x1,x2},{x1}},x3},{{{x1,x2},{x1}}}}}},x5},{{{{{{{x1,x2},{x1}},x3},{{{x1,x2},{x1}}}},x4},{{{{{x1,x2},{x1}},x3},{{{x1,x2},{x1}}}}}}}}}} by TARSKI:def 5 ;
then A8: {{{{{{{{{x1,x2},{x1}},x3},{{{x1,x2},{x1}}}},x4},{{{{{x1,x2},{x1}},x3},{{{x1,x2},{x1}}}}}},x5},{{{{{{{x1,x2},{x1}},x3},{{{x1,x2},{x1}}}},x4},{{{{{x1,x2},{x1}},x3},{{{x1,x2},{x1}}}}}}}},x6} in Y by TARSKI:def 2;
A9: ( {{{{{x1,x2},{x1}},x3},{{{x1,x2},{x1}}}},x4} in {{{{{{x1,x2},{x1}},x3},{{{x1,x2},{x1}}}},x4},{{{{{x1,x2},{x1}},x3},{{{x1,x2},{x1}}}}}} & {{{{{{x1,x2},{x1}},x3},{{{x1,x2},{x1}}}},x4},{{{{{x1,x2},{x1}},x3},{{{x1,x2},{x1}}}}}} in {{{{{{{x1,x2},{x1}},x3},{{{x1,x2},{x1}}}},x4},{{{{{x1,x2},{x1}},x3},{{{x1,x2},{x1}}}}}},x5} ) by TARSKI:def 2;
( {x1,x2} in {{x1,x2},{x1}} & {{x1,x2},{x1}} in {{{x1,x2},{x1}},x3} ) by TARSKI:def 2;
hence contradiction by A2, A6, A5, A9, A7, A8; :: thesis: verum