Theorem prssg 3737

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 3716	. . 3 |- ( A e. V -> ( A e. C <-> { A } C_ C ) )
2		snssg 3716	. . 3 |- ( B e. W -> ( B e. C <-> { B } C_ C ) )
3	1, 2	bi2anan9 601	. 2 |- ( ( A e. V /\ B e. W ) -> ( ( A e. C /\ B e. C ) <-> ( { A } C_ C /\ { B } C_ C ) ) )
4		unss 3301	. . 3 |- ( ( { A } C_ C /\ { B } C_ C ) <-> ( { A } u. { B } ) C_ C )
5		df-pr 3590	. . . 4 |- { A , B } = ( { A } u. { B } )
6	5	sseq1i 3173	. . 3 |- ( { A , B } C_ C <-> ( { A } u. { B } ) C_ C )
7	4, 6	bitr4i 186	. 2 |- ( ( { A } C_ C /\ { B } C_ C ) <-> { A , B } C_ C )
8	3, 7	bitrdi 195	1 |- ( ( A e. V /\ B e. W ) -> ( ( A e. C /\ B e. C ) <-> { A , B } C_ C ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   e. wcel 2141   u. cun 3119   C_ wss 3121   {csn 3583   {cpr 3584

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-sn 3589  df-pr 3590

This theorem is referenced by:  prssi  3738  prsspwg  3739  topgele  12821