let A, B be set ; ( ( for x being set holds
( x in A iff x is bag of X ) ) & ( for x being set holds
( x in B iff x is bag of X ) ) implies A = B )
assume that
A3:
for x being set holds
( x in A iff x is bag of X )
and
A4:
for x being set holds
( x in B iff x is bag of X )
; A = B
hence
A = B
by TARSKI:2; verum