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Metamath Proof Explorer

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 = A -> ( x e. B <-> A e. B ) )
5 eleq1 
 |-  ( x = A -> ( x e. C <-> A e. C ) )
6 4 5 anbi12d 
 |-  ( x = A -> ( ( x e. B /\ x e. C ) <-> ( A e. B /\ A e. C ) ) )
7 df-in 
 |-  ( B i^i C ) = { x | ( x e. B /\ x e. C ) }
8 6 7 elab2g 
 |-  ( A e. _V -> ( A e. ( B i^i C ) <-> ( A e. B /\ A e. C ) ) )
9 1 3 8 pm5.21nii 
 |-  ( A e. ( B i^i C ) <-> ( A e. B /\ A e. C ) )