let X be set ; :: thesis: ( X <> {} implies ex v being object st
( v in X & ( for x1, x2, x3, x4 being object holds
( ( not x1 in X & not x2 in X ) or not v = [x1,x2,x3,x4] ) ) ) )
assume X <> {} ; :: thesis: ex v being object st
( v in X & ( for x1, x2, x3, x4 being object holds
( ( not x1 in X & not x2 in X ) or not v = [x1,x2,x3,x4] ) ) )
then consider Y being set such that
A1: Y in X and
A2: for Y1, Y2, Y3, Y4, Y5 being set st Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y holds
Y1 misses X by XREGULAR:6;
take v = Y; :: thesis: ( v in X & ( for x1, x2, x3, x4 being object holds
( ( not x1 in X & not x2 in X ) or not v = [x1,x2,x3,x4] ) ) )
thus v in X by A1; :: thesis: for x1, x2, x3, x4 being object holds
( ( not x1 in X & not x2 in X ) or not v = [x1,x2,x3,x4] )
given x1, x2, x3, x4 being object such that A3: ( x1 in X or x2 in X ) and
A4: v = [x1,x2,x3,x4] ; :: 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};
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;
A6: {{{{{x1,x2},{x1}},x3},{{{x1,x2},{x1}}}},x4} in Y by A4, TARSKI:def 2;
A7: ( x1 in {x1,x2} & x2 in {x1,x2} ) 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, A7, A5, A6, A3, XBOOLE_0:3; :: thesis: verum