Theorem elprg

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, generalized. (Contributed by NM, 13-Sep-1995)

	Ref	Expression
Assertion	elprg	|- ( A e. V -> ( A e. { B , C } <-> ( A = B \/ A = C ) ) )

Proof

Step	Hyp	Ref	Expression
1		eqeq1	|- ( x = y -> ( x = B <-> y = B ) )
2		eqeq1	|- ( x = y -> ( x = C <-> y = C ) )
3	1 2	orbi12d	|- ( x = y -> ( ( x = B \/ x = C ) <-> ( y = B \/ y = C ) ) )
4		eqeq1	|- ( y = A -> ( y = B <-> A = B ) )
5		eqeq1	|- ( y = A -> ( y = C <-> A = C ) )
6	4 5	orbi12d	|- ( y = A -> ( ( y = B \/ y = C ) <-> ( A = B \/ A = C ) ) )
7		dfpr2	|- { B , C } = { x | ( x = B \/ x = C ) }
8	3 6 7	elab2gw	|- ( A e. V -> ( A e. { B , C } <-> ( A = B \/ A = C ) ) )