Theorem prssg 3830

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 3807	. . 3 |- ( A e. V -> ( A e. C <-> { A } C_ C ) )
2		snssg 3807	. . 3 |- ( B e. W -> ( B e. C <-> { B } C_ C ) )
3	1, 2	bi2anan9 610	. 2 |- ( ( A e. V /\ B e. W ) -> ( ( A e. C /\ B e. C ) <-> ( { A } C_ C /\ { B } C_ C ) ) )
4		unss 3381	. . 3 |- ( ( { A } C_ C /\ { B } C_ C ) <-> ( { A } u. { B } ) C_ C )
5		df-pr 3676	. . . 4 |- { A , B } = ( { A } u. { B } )
6	5	sseq1i 3253	. . 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 2202   u. cun 3198   C_ wss 3200   {csn 3669   {cpr 3670

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-sn 3675  df-pr 3676

This theorem is referenced by:  prssi  3831  prsspwg  3833  ssprss  3834  prelpw  4305  hashdmprop2dom  11107  topgele  14752  structgrssvtx  15892  structgrssiedg  15893  umgredgprv  15965  wlk1walkdom  16209