Theorem prsspw 3557

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 3541	. 2 |- ( ( A e. ~P C /\ B e. ~P C ) <-> { A , B } C_ ~P C )
4	1	elpw 3388	. . 3 |- ( A e. ~P C <-> A C_ C )
5	2	elpw 3388	. . 3 |- ( B e. ~P C <-> B C_ C )
6	4, 5	anbi12i 447	. 2 |- ( ( A e. ~P C /\ B e. ~P C ) <-> ( A C_ C /\ B C_ C ) )
7	3, 6	bitr3i 184	1 |- ( { A , B } C_ ~P C <-> ( A C_ C /\ B C_ C ) )

Syntax hints:   /\ wa 102   <-> wb 103   e. wcel 1433   _Vcvv 2601   C_ wss 2973   ~Pcpw 3382   {cpr 3399

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405

This theorem is referenced by: (None)