Theorem prsspw 3853

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 3834	. 2 |- ( ( A e. ~P C /\ B e. ~P C ) <-> { A , B } C_ ~P C )
4	1	elpw 3662	. . 3 |- ( A e. ~P C <-> A C_ C )
5	2	elpw 3662	. . 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 2202   _Vcvv 2803   C_ wss 3201   ~Pcpw 3656   {cpr 3674

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680

This theorem is referenced by: (None)