Theorem prss 3850

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 3393	. 2 |- ( ( { A } C_ C /\ { B } C_ C ) <-> ( { A } u. { B } ) C_ C )
2		prss.1	. . . 4 |- A e. _V
3	2	snss 3829	. . 3 |- ( A e. C <-> { A } C_ C )
4		prss.2	. . . 4 |- B e. _V
5	4	snss 3829	. . 3 |- ( B e. C <-> { B } C_ C )
6	3, 5	anbi12i 460	. 2 |- ( ( A e. C /\ B e. C ) <-> ( { A } C_ C /\ { B } C_ C ) )
7		df-pr 3696	. . 3 |- { A , B } = ( { A } u. { B } )
8	7	sseq1i 3264	. 2 |- ( { A , B } C_ C <-> ( { A } u. { B } ) C_ C )
9	1, 6, 8	3bitr4i 212	1 |- ( ( A e. C /\ B e. C ) <-> { A , B } C_ C )

Syntax hints:   /\ wa 104   <-> wb 105   e. wcel 2203   _Vcvv 2813   u. cun 3209   C_ wss 3211   {csn 3689   {cpr 3690

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2815  df-un 3215  df-in 3217  df-ss 3224  df-sn 3695  df-pr 3696

This theorem is referenced by:  tpss  3862  prsspw  3869  exmidpw  7168  pw1ne1  7539  prdsex  13482  prdsval  13486  prdsbaslemss  13487  releqgg  13937  eqgex  13938  eqgfval  13939  eqgval  13940  umgredg  16140