Theorem prelpwi 4243

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 3776	. 2 |- ( ( A e. C /\ B e. C ) -> { A , B } C_ C )
2		prexg 4240	. . 3 |- ( ( A e. C /\ B e. C ) -> { A , B } e. _V )
3		elpwg 3609	. . 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 2164   _Vcvv 2760   C_ wss 3153   ~Pcpw 3601   {cpr 3619

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625

This theorem is referenced by: (None)