Theorem prelpwi 3950

Description: A pair of two sets belongs to the power class of a class containing those two sets. (Contributed by Thierry Arnoux, 10-Mar-2017.)

Assertion

Ref	Expression
prelpwi	|- ( ( A e. C /\ B e. C ) -> { A , B } e. ~P C )

Proof of Theorem prelpwi

Step	Hyp	Ref	Expression
1		prssi 3522	. 2 |- ( ( A e. C /\ B e. C ) -> { A , B } C_ C )
2		elex 2566	. . . 4 |- ( A e. C -> A e. _V )
3		elex 2566	. . . 4 |- ( B e. C -> B e. _V )
4		prexgOLD 3946	. . . 4 |- ( ( A e. _V /\ B e. _V ) -> { A , B } e. _V )
5	2, 3, 4	syl2an 273	. . 3 |- ( ( A e. C /\ B e. C ) -> { A , B } e. _V )
6		elpwg 3367	. . 3 |- ( { A , B } e. _V -> ( { A , B } e. ~P C <-> { A , B } C_ C ) )
7	5, 6	syl 14	. 2 |- ( ( A e. C /\ B e. C ) -> ( { A , B } e. ~P C <-> { A , B } C_ C ) )
8	1, 7	mpbird 156	1 |- ( ( A e. C /\ B e. C ) -> { A , B } e. ~P C )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   e. wcel 1393   _Vcvv 2557   C_ wss 2917   ~Pcpw 3359   {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-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pr 3944

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-pw 3361  df-sn 3381  df-pr 3382

This theorem is referenced by: (None)