Theorem prsspw 3767

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.)

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	. . 3 |- A e. _V
2		prsspw.2	. . 3 |- B e. _V
3	1, 2	prss 3750	. 2 |- ( ( A e. ~P C /\ B e. ~P C ) <-> { A , B } C_ ~P C )
4	1	elpw 3583	. . 3 |- ( A e. ~P C <-> A C_ C )
5	2	elpw 3583	. . 3 |- ( B e. ~P C <-> B C_ C )
6	4, 5	anbi12i 460	. 2 |- ( ( A e. ~P C /\ B e. ~P C ) <-> ( A C_ C /\ B C_ C ) )
7	3, 6	bitr3i 186	1 |- ( { A , B } C_ ~P C <-> ( A C_ C /\ B C_ C ) )

Syntax hints:   /\ wa 104   <-> wb 105   e. wcel 2148   _Vcvv 2739   C_ wss 3131   ~Pcpw 3577   {cpr 3595

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601

This theorem is referenced by: (None)