Theorem prssg 3672

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 3651	. . 3 |- ( A e. V -> ( A e. C <-> { A } C_ C ) )
2		snssg 3651	. . 3 |- ( B e. W -> ( B e. C <-> { B } C_ C ) )
3	1, 2	bi2anan9 595	. 2 |- ( ( A e. V /\ B e. W ) -> ( ( A e. C /\ B e. C ) <-> ( { A } C_ C /\ { B } C_ C ) ) )
4		unss 3245	. . 3 |- ( ( { A } C_ C /\ { B } C_ C ) <-> ( { A } u. { B } ) C_ C )
5		df-pr 3529	. . . 4 |- { A , B } = ( { A } u. { B } )
6	5	sseq1i 3118	. . 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 1480   u. cun 3064   C_ wss 3066   {csn 3522   {cpr 3523

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-sn 3528  df-pr 3529

This theorem is referenced by:  prssi  3673  prsspwg  3674  topgele  12185