Theorem prss 3736

Description: A pair of elements of a class is a subset of the class. Theorem 7.5 of [Quine] p. 49. (Contributed by NM, 30-May-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Hypotheses

Ref	Expression
prss.1	|- A e. _V
prss.2	|- B e. _V

Assertion

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

Proof of Theorem prss

Step	Hyp	Ref	Expression
1		unss 3301	. 2 |- ( ( { A } C_ C /\ { B } C_ C ) <-> ( { A } u. { B } ) C_ C )
2		prss.1	. . . 4 |- A e. _V
3	2	snss 3709	. . 3 |- ( A e. C <-> { A } C_ C )
4		prss.2	. . . 4 |- B e. _V
5	4	snss 3709	. . 3 |- ( B e. C <-> { B } C_ C )
6	3, 5	anbi12i 457	. 2 |- ( ( A e. C /\ B e. C ) <-> ( { A } C_ C /\ { B } C_ C ) )
7		df-pr 3590	. . 3 |- { A , B } = ( { A } u. { B } )
8	7	sseq1i 3173	. 2 |- ( { A , B } C_ C <-> ( { A } u. { B } ) C_ C )
9	1, 6, 8	3bitr4i 211	1 |- ( ( A e. C /\ B e. C ) <-> { A , B } C_ C )

Syntax hints:   /\ wa 103   <-> wb 104   e. wcel 2141   _Vcvv 2730   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:  tpss  3745  prsspw  3752  exmidpw  6886  pw1ne1  7206