Theorem prelpw 4298

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 3824	. 2 |- ( ( A e. V /\ B e. W ) -> ( ( A e. C /\ B e. C ) <-> { A , B } C_ C ) )
2		prexg 4294	. . 3 |- ( ( A e. V /\ B e. W ) -> { A , B } e. _V )
3		elpwg 3657	. . 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 2200   _Vcvv 2799   C_ wss 3197   ~Pcpw 3649   {cpr 3667

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pr 4292

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673

This theorem is referenced by:  umgrpredgv  15939