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Theorem prsspw 4043
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 4025 . 2  |-  ( ( A  e.  ~P C  /\  B  e.  ~P C )  <->  { A ,  B }  C_  ~P C )
41elpw 3864 . . 3  |-  ( A  e.  ~P C  <->  A  C_  C
)
52elpw 3864 . . 3  |-  ( B  e.  ~P C  <->  B  C_  C
)
64, 5anbi12i 697 . 2  |-  ( ( A  e.  ~P C  /\  B  e.  ~P C )  <->  ( A  C_  C  /\  B  C_  C ) )
73, 6bitr3i 251 1  |-  ( { A ,  B }  C_ 
~P C  <->  ( A  C_  C  /\  B  C_  C ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    e. wcel 1756   _Vcvv 2970    C_ wss 3326   ~Pcpw 3858   {cpr 3877
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-v 2972  df-un 3331  df-in 3333  df-ss 3340  df-pw 3860  df-sn 3876  df-pr 3878
This theorem is referenced by:  altxpsspw  28006
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