Theorem prssg 3643

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 3622	. . 3 |- ( A e. V -> ( A e. C <-> { A } C_ C ) )
2		snssg 3622	. . 3 |- ( B e. W -> ( B e. C <-> { B } C_ C ) )
3	1, 2	bi2anan9 578	. 2 |- ( ( A e. V /\ B e. W ) -> ( ( A e. C /\ B e. C ) <-> ( { A } C_ C /\ { B } C_ C ) ) )
4		unss 3216	. . 3 |- ( ( { A } C_ C /\ { B } C_ C ) <-> ( { A } u. { B } ) C_ C )
5		df-pr 3500	. . . 4 |- { A , B } = ( { A } u. { B } )
6	5	sseq1i 3089	. . 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	syl6bb 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 1463   u. cun 3035   C_ wss 3037   {csn 3493   {cpr 3494

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097

This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-v 2659  df-un 3041  df-in 3043  df-ss 3050  df-sn 3499  df-pr 3500

This theorem is referenced by:  prssi  3644  prsspwg  3645  topgele  12036