Theorem prssg 3779

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.)

Assertion

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

Proof of Theorem prssg

Step	Hyp	Ref	Expression
1		snssg 3756	. . 3 |- ( A e. V -> ( A e. C <-> { A } C_ C ) )
2		snssg 3756	. . 3 |- ( B e. W -> ( B e. C <-> { B } C_ C ) )
3	1, 2	bi2anan9 606	. 2 |- ( ( A e. V /\ B e. W ) -> ( ( A e. C /\ B e. C ) <-> ( { A } C_ C /\ { B } C_ C ) ) )
4		unss 3337	. . 3 |- ( ( { A } C_ C /\ { B } C_ C ) <-> ( { A } u. { B } ) C_ C )
5		df-pr 3629	. . . 4 |- { A , B } = ( { A } u. { B } )
6	5	sseq1i 3209	. . 3 |- ( { A , B } C_ C <-> ( { A } u. { B } ) C_ C )
7	4, 6	bitr4i 187	. 2 |- ( ( { A } C_ C /\ { B } C_ C ) <-> { A , B } C_ C )
8	3, 7	bitrdi 196	1 |- ( ( A e. V /\ B e. W ) -> ( ( A e. C /\ B e. C ) <-> { A , B } C_ C ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   e. wcel 2167   u. cun 3155   C_ wss 3157   {csn 3622   {cpr 3623

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-sn 3628  df-pr 3629

This theorem is referenced by:  prssi  3780  prsspwg  3782  topgele  14265