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Theorem prsspw 3973
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
StepHypRef Expression
1 prsspw.1 . . 3  |-  A  e. 
_V
2 prsspw.2 . . 3  |-  B  e. 
_V
31, 2prss 3954 . 2  |-  ( ( A  e.  ~P C  /\  B  e.  ~P C )  <->  { A ,  B }  C_  ~P C )
41elpw 3807 . . 3  |-  ( A  e.  ~P C  <->  A  C_  C
)
52elpw 3807 . . 3  |-  ( B  e.  ~P C  <->  B  C_  C
)
64, 5anbi12i 680 . 2  |-  ( ( A  e.  ~P C  /\  B  e.  ~P C )  <->  ( A  C_  C  /\  B  C_  C ) )
73, 6bitr3i 244 1  |-  ( { A ,  B }  C_ 
~P C  <->  ( A  C_  C  /\  B  C_  C ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360    e. wcel 1726   _Vcvv 2958    C_ wss 3322   ~Pcpw 3801   {cpr 3817
This theorem is referenced by:  altxpsspw  25824
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-v 2960  df-un 3327  df-in 3329  df-ss 3336  df-pw 3803  df-sn 3822  df-pr 3823
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