let X be set ; :: thesis: singletons X misses 2Set X
assume singletons X meets 2Set X ; :: thesis: contradiction
then consider x being object such that
A1: x in (singletons X) /\ (2Set X) by XBOOLE_0:4;
A2: ( x in singletons X & x in 2Set X ) by A1, XBOOLE_0:def 4;
then consider Y being Subset of X such that
A3: ( x = Y & Y is 1 -element ) ;
consider Z being Subset of X such that
A4: ( x = Z & card Z = 2 ) by A2;
thus contradiction by A3, A4, CARD_1:def 7; :: thesis: verum