Theorem prelpw 4329

Description: An unordered pair of two sets is a member of the powerclass of a class if and only if the two sets are members of that class. (Contributed by AV, 8-Jan-2020.)

Assertion

Ref	Expression
prelpw	|- ( ( A e. V /\ B e. W ) -> ( ( A e. C /\ B e. C ) <-> { A , B } e. ~P C ) )

Proof of Theorem prelpw

Step	Hyp	Ref	Expression
1		prssg 3851	. 2 |- ( ( A e. V /\ B e. W ) -> ( ( A e. C /\ B e. C ) <-> { A , B } C_ C ) )
2		prexg 4325	. . 3 |- ( ( A e. V /\ B e. W ) -> { A , B } e. _V )
3		elpwg 3677	. . 3 |- ( { A , B } e. _V -> ( { A , B } e. ~P C <-> { A , B } C_ C ) )
4	2, 3	syl 14	. 2 |- ( ( A e. V /\ B e. W ) -> ( { A , B } e. ~P C <-> { A , B } C_ C ) )
5	1, 4	bitr4d 191	1 |- ( ( A e. V /\ B e. W ) -> ( ( A e. C /\ B e. C ) <-> { A , B } e. ~P C ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   e. wcel 2203   _Vcvv 2813   C_ wss 3211   ~Pcpw 3669   {cpr 3690

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 2206  ax-ext 2214  ax-sep 4228  ax-pr 4322

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2815  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696

This theorem is referenced by:  umgrpredgv  16142