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Metamath Proof Explorer

Theorem elex22

Description: If two classes each contain another class, then both contain some set. (Contributed by Alan Sare, 24-Oct-2011)

Ref Expression
Assertion elex22 
|- ( ( A e. B /\ A e. C ) -> E. x ( x e. B /\ x e. C ) )

Proof

Step Hyp Ref Expression
1 eleq1a 
 |-  ( A e. B -> ( x = A -> x e. B ) )
2 eleq1a 
 |-  ( A e. C -> ( x = A -> x e. C ) )
3 1 2 anim12ii 
 |-  ( ( A e. B /\ A e. C ) -> ( x = A -> ( x e. B /\ x e. C ) ) )
4 3 alrimiv 
 |-  ( ( A e. B /\ A e. C ) -> A. x ( x = A -> ( x e. B /\ x e. C ) ) )
5 elissetv 
 |-  ( A e. B -> E. x x = A )
6 5 adantr 
 |-  ( ( A e. B /\ A e. C ) -> E. x x = A )
7 exim 
 |-  ( A. x ( x = A -> ( x e. B /\ x e. C ) ) -> ( E. x x = A -> E. x ( x e. B /\ x e. C ) ) )
8 4 6 7 sylc 
 |-  ( ( A e. B /\ A e. C ) -> E. x ( x e. B /\ x e. C ) )