Theorem prss 3541

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 3146	. 2 |- ( ( { A } C_ C /\ { B } C_ C ) <-> ( { A } u. { B } ) C_ C )
2		prss.1	. . . 4 |- A e. _V
3	2	snss 3516	. . 3 |- ( A e. C <-> { A } C_ C )
4		prss.2	. . . 4 |- B e. _V
5	4	snss 3516	. . 3 |- ( B e. C <-> { B } C_ C )
6	3, 5	anbi12i 447	. 2 |- ( ( A e. C /\ B e. C ) <-> ( { A } C_ C /\ { B } C_ C ) )
7		df-pr 3405	. . 3 |- { A , B } = ( { A } u. { B } )
8	7	sseq1i 3023	. 2 |- ( { A , B } C_ C <-> ( { A } u. { B } ) C_ C )
9	1, 6, 8	3bitr4i 210	1 |- ( ( A e. C /\ B e. C ) <-> { A , B } C_ C )

Syntax hints:   /\ wa 102   <-> wb 103   e. wcel 1433   _Vcvv 2601   u. cun 2971   C_ wss 2973   {csn 3398   {cpr 3399

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-sn 3404  df-pr 3405

This theorem is referenced by:  tpss  3550  prsspw  3557