Theorem prsspw 3752

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 3736	. 2 |- ( ( A e. ~P C /\ B e. ~P C ) <-> { A , B } C_ ~P C )
4	1	elpw 3572	. . 3 |- ( A e. ~P C <-> A C_ C )
5	2	elpw 3572	. . 3 |- ( B e. ~P C <-> B C_ C )
6	4, 5	anbi12i 457	. 2 |- ( ( A e. ~P C /\ B e. ~P C ) <-> ( A C_ C /\ B C_ C ) )
7	3, 6	bitr3i 185	1 |- ( { A , B } C_ ~P C <-> ( A C_ C /\ B C_ C ) )

Syntax hints:   /\ wa 103   <-> wb 104   e. wcel 2141   _Vcvv 2730   C_ wss 3121   ~Pcpw 3566   {cpr 3584

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590

This theorem is referenced by: (None)