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Metamath Proof Explorer

Theorem elun

Description: Expansion of membership in class union. Theorem 12 of Suppes p. 25. (Contributed by NM, 7-Aug-1994)

Ref Expression
Assertion elun 
|- ( A e. ( B u. C ) <-> ( A e. B \/ A e. C ) )

Proof

Step Hyp Ref Expression
1 elex 
 |-  ( A e. ( B u. C ) -> A e. _V )
2 elex 
 |-  ( A e. B -> A e. _V )
3 elex 
 |-  ( A e. C -> A e. _V )
4 2 3 jaoi 
 |-  ( ( A e. B \/ A e. C ) -> A e. _V )
5 eleq1 
 |-  ( x = A -> ( x e. B <-> A e. B ) )
6 eleq1 
 |-  ( x = A -> ( x e. C <-> A e. C ) )
7 5 6 orbi12d 
 |-  ( x = A -> ( ( x e. B \/ x e. C ) <-> ( A e. B \/ A e. C ) ) )
8 df-un 
 |-  ( B u. C ) = { x | ( x e. B \/ x e. C ) }
9 7 8 elab2g 
 |-  ( A e. _V -> ( A e. ( B u. C ) <-> ( A e. B \/ A e. C ) ) )
10 1 4 9 pm5.21nii 
 |-  ( A e. ( B u. C ) <-> ( A e. B \/ A e. C ) )