Theorem prss 3834

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 3383	. 2 |- ( ( { A } C_ C /\ { B } C_ C ) <-> ( { A } u. { B } ) C_ C )
2		prss.1	. . . 4 |- A e. _V
3	2	snss 3813	. . 3 |- ( A e. C <-> { A } C_ C )
4		prss.2	. . . 4 |- B e. _V
5	4	snss 3813	. . 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 3680	. . 3 |- { A , B } = ( { A } u. { B } )
8	7	sseq1i 3254	. 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 2202   _Vcvv 2803   u. cun 3199   C_ wss 3201   {csn 3673   {cpr 3674

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 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-sn 3679  df-pr 3680

This theorem is referenced by:  tpss  3846  prsspw  3853  exmidpw  7143  pw1ne1  7507  prdsex  13432  prdsval  13436  prdsbaslemss  13437  releqgg  13887  eqgex  13888  eqgfval  13889  eqgval  13890  umgredg  16086