Theorem prsspwg 3804

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 3801	. 2 |- ( ( A e. V /\ B e. W ) -> ( ( A e. ~P C /\ B e. ~P C ) <-> { A , B } C_ ~P C ) )
2		elpwg 3634	. . 3 |- ( A e. V -> ( A e. ~P C <-> A C_ C ) )
3		elpwg 3634	. . 3 |- ( B e. W -> ( B e. ~P C <-> B C_ C ) )
4	2, 3	bi2anan9 606	. 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 190	1 |- ( ( A e. V /\ B e. W ) -> ( { A , B } C_ ~P C <-> ( A C_ C /\ B C_ C ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   e. wcel 2178   C_ wss 3174   ~Pcpw 3626   {cpr 3644

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650

This theorem is referenced by: (None)