Theorem prelpwi 4131

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 3673	. 2 |- ( ( A e. C /\ B e. C ) -> { A , B } C_ C )
2		prexg 4128	. . 3 |- ( ( A e. C /\ B e. C ) -> { A , B } e. _V )
3		elpwg 3513	. . 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 166	1 |- ( ( A e. C /\ B e. C ) -> { A , B } e. ~P C )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   e. wcel 1480   _Vcvv 2681   C_ wss 3066   ~Pcpw 3505   {cpr 3523

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pr 4126

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529

This theorem is referenced by: (None)