let X, Y be set ; X /\ (Y --> X) = {}
assume
X /\ (Y --> X) <> {}
; contradiction
then consider Z being object such that
A1:
Z in X /\ (Y --> X)
by XBOOLE_0:def 1;
reconsider Z = Z as set by TARSKI:1;
consider y, x being object such that
A2:
( y in Y & x in {X} & Z = [y,x] )
by ZFMISC_1:def 2, A1;
x = X
by A2, TARSKI:def 1;
then
Z = {{y,X},{y}}
by A2, TARSKI:def 5;
then A3:
{y,X} in Z
by TARSKI:def 2;
( X in {y,X} & Z in X )
by A1, XBOOLE_0:def 4, TARSKI:def 2;
hence
contradiction
by A3, XREGULAR:7; verum