Theorem elin

Description: Expansion of membership in an intersection of two classes. Theorem 12 of Suppes p. 25. (Contributed by NM, 29-Apr-1994)

	Ref	Expression
Assertion	elin	|- ( A e. ( B i^i C ) <-> ( A e. B /\ A e. C ) )

Proof

Step	Hyp	Ref	Expression
1		elex	|- ( A e. ( B i^i C ) -> A e. _V )
2		elex	|- ( A e. C -> A e. _V )
3	2	adantl	|- ( ( A e. B /\ A e. C ) -> A e. _V )
4		eleq1	|- ( x = y -> ( x e. B <-> y e. B ) )
5		eleq1	|- ( x = y -> ( x e. C <-> y e. C ) )
6	4 5	anbi12d	|- ( x = y -> ( ( x e. B /\ x e. C ) <-> ( y e. B /\ y e. C ) ) )
7		eleq1	|- ( y = A -> ( y e. B <-> A e. B ) )
8		eleq1	|- ( y = A -> ( y e. C <-> A e. C ) )
9	7 8	anbi12d	|- ( y = A -> ( ( y e. B /\ y e. C ) <-> ( A e. B /\ A e. C ) ) )
10		df-in	|- ( B i^i C ) = { x | ( x e. B /\ x e. C ) }
11	6 9 10	elab2gw	|- ( A e. _V -> ( A e. ( B i^i C ) <-> ( A e. B /\ A e. C ) ) )
12	1 3 11	pm5.21nii	|- ( A e. ( B i^i C ) <-> ( A e. B /\ A e. C ) )