Theorem prsspw 4376

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.)

Hypotheses

Ref	Expression
prsspw.1	|- A e. _V
prsspw.2	|- B e. _V

Assertion

Ref	Expression
prsspw	|- ( { A , B } C_ ~P C <-> ( A C_ C /\ B C_ C ) )

Proof of Theorem prsspw

Step	Hyp	Ref	Expression
1		prsspw.1	. 2 |- A e. _V
2		prsspw.2	. 2 |- B e. _V
3		prsspwg 4355	. 2 |- ( ( A e. _V /\ B e. _V ) -> ( { A , B } C_ ~P C <-> ( A C_ C /\ B C_ C ) ) )
4	1, 2, 3	mp2an 708	1 |- ( { A , B } C_ ~P C <-> ( A C_ C /\ B C_ C ) )

Syntax hints:   <-> wb 196   /\ wa 384   e. wcel 1990   _Vcvv 3200   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:  altxpsspw  32084