Theorem elpr

Description: A member of a pair of classes is one or the other of them, and conversely as soon as it is a set. Exercise 1 of TakeutiZaring p. 15. (Contributed by NM, 13-Sep-1995)

	Ref	Expression
Hypothesis	elpr.1	|- A e. _V
Assertion	elpr	|- ( A e. { B , C } <-> ( A = B \/ A = C ) )

Proof

Step	Hyp	Ref	Expression
1		elpr.1	|- A e. _V
2		elprg	|- ( A e. _V -> ( A e. { B , C } <-> ( A = B \/ A = C ) ) )
3	1 2	ax-mp	|- ( A e. { B , C } <-> ( A = B \/ A = C ) )