Theorem prelpwi 4192

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 3731	. 2 |- ( ( A e. C /\ B e. C ) -> { A , B } C_ C )
2		prexg 4189	. . 3 |- ( ( A e. C /\ B e. C ) -> { A , B } e. _V )
3		elpwg 3567	. . 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 2136   _Vcvv 2726   C_ wss 3116   ~Pcpw 3559   {cpr 3577

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583

This theorem is referenced by: (None)