Theorem prss 3520

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 3117	. 2 |- ( ( { A } C_ C /\ { B } C_ C ) <-> ( { A } u. { B } ) C_ C )
2		prss.1	. . . 4 |- A e. _V
3	2	snss 3494	. . 3 |- ( A e. C <-> { A } C_ C )
4		prss.2	. . . 4 |- B e. _V
5	4	snss 3494	. . 3 |- ( B e. C <-> { B } C_ C )
6	3, 5	anbi12i 433	. 2 |- ( ( A e. C /\ B e. C ) <-> ( { A } C_ C /\ { B } C_ C ) )
7		df-pr 3382	. . 3 |- { A , B } = ( { A } u. { B } )
8	7	sseq1i 2969	. 2 |- ( { A , B } C_ C <-> ( { A } u. { B } ) C_ C )
9	1, 6, 8	3bitr4i 201	1 |- ( ( A e. C /\ B e. C ) <-> { A , B } C_ C )

Syntax hints:   /\ wa 97   <-> wb 98   e. wcel 1393   _Vcvv 2557   u. cun 2915   C_ wss 2917   {csn 3375   {cpr 3376

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-sn 3381  df-pr 3382

This theorem is referenced by:  tpss  3529  prsspw  3536