Theorem prsspw

Description: An unordered pair belongs to the power class of a class iff each member belongs to the class. (Contributed by NM, 10-Dec-2003) (Proof shortened by Andrew Salmon, 26-Jun-2011) (Proof shortened by OpenAI, 25-Mar-2020)

	Ref	Expression
Hypotheses	prsspw.1	|- A e. _V
	prsspw.2	|- B e. _V
Assertion	prsspw	|- ( { A , B } C_ ~P C <-> ( A C_ C /\ B C_ C ) )

Proof

Step	Hyp	Ref	Expression
1		prsspw.1	|- A e. _V
2		prsspw.2	|- B e. _V
3		prsspwg	|- ( ( A e. _V /\ B e. _V ) -> ( { A , B } C_ ~P C <-> ( A C_ C /\ B C_ C ) ) )
4	1 2 3	mp2an	|- ( { A , B } C_ ~P C <-> ( A C_ C /\ B C_ C ) )