Theorem prsspwg 4355

Description: An unordered pair belongs to the power class of a class iff each member belongs to the class. (Contributed by Thierry Arnoux, 3-Oct-2016.) (Revised by NM, 18-Jan-2018.)

Assertion

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

Proof of Theorem prsspwg

Step	Hyp	Ref	Expression
1		prssg 4350	. 2 |- ( ( A e. V /\ B e. W ) -> ( ( A e. ~P C /\ B e. ~P C ) <-> { A , B } C_ ~P C ) )
2		elpwg 4166	. . 3 |- ( A e. V -> ( A e. ~P C <-> A C_ C ) )
3		elpwg 4166	. . 3 |- ( B e. W -> ( B e. ~P C <-> B C_ C ) )
4	2, 3	bi2anan9 917	. 2 |- ( ( A e. V /\ B e. W ) -> ( ( A e. ~P C /\ B e. ~P C ) <-> ( A C_ C /\ B C_ C ) ) )
5	1, 4	bitr3d 270	1 |- ( ( A e. V /\ B e. W ) -> ( { A , B } C_ ~P C <-> ( A C_ C /\ B C_ C ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   e. wcel 1990   C_ wss 3574   ~Pcpw 4158   {cpr 4179

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-un 3579  df-in 3581  df-ss 3588  df-pw 4160  df-sn 4178  df-pr 4180

This theorem is referenced by:  prsspw  4376