Theorem prsspwg

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)

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

Proof

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