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Metamath Proof Explorer

Theorem prssg

Description: A pair of elements of a class is a subset of the class. Theorem 7.5 of Quine p. 49. (Contributed by NM, 22-Mar-2006) (Proof shortened by Andrew Salmon, 29-Jun-2011)

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

Proof

Step Hyp Ref Expression
1 snssg 
 |-  ( A e. V -> ( A e. C <-> { A } C_ C ) )
2 snssg 
 |-  ( B e. W -> ( B e. C <-> { B } C_ C ) )
3 1 2 bi2anan9 
 |-  ( ( A e. V /\ B e. W ) -> ( ( A e. C /\ B e. C ) <-> ( { A } C_ C /\ { B } C_ C ) ) )
4 unss 
 |-  ( ( { A } C_ C /\ { B } C_ C ) <-> ( { A } u. { B } ) C_ C )
5 df-pr 
 |-  { A , B } = ( { A } u. { B } )
6 5 sseq1i 
 |-  ( { A , B } C_ C <-> ( { A } u. { B } ) C_ C )
7 4 6 bitr4i 
 |-  ( ( { A } C_ C /\ { B } C_ C ) <-> { A , B } C_ C )
8 3 7 bitrdi 
 |-  ( ( A e. V /\ B e. W ) -> ( ( A e. C /\ B e. C ) <-> { A , B } C_ C ) )