Theorem prelpwi 4335

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 3857	. 2 |- ( ( A e. C /\ B e. C ) -> { A , B } C_ C )
2		prexg 4330	. . 3 |- ( ( A e. C /\ B e. C ) -> { A , B } e. _V )
3		elpwg 3682	. . 3 |- ( { A , B } e. _V -> ( { A , B } e. ~P C <-> { A , B } C_ C ) )
4	2, 3	syl 14	. 2 |- ( ( A e. C /\ B e. C ) -> ( { A , B } e. ~P C <-> { A , B } C_ C ) )
5	1, 4	mpbird 167	1 |- ( ( A e. C /\ B e. C ) -> { A , B } e. ~P C )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   e. wcel 2205   _Vcvv 2815   C_ wss 3214   ~Pcpw 3674   {cpr 3695

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-14 2208  ax-ext 2216  ax-sep 4233  ax-pr 4327

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701

This theorem is referenced by:  vdegp1aid  16435  vdegp1bid  16436  eupth2lem3lem5  16593  konigsberglem1  16609