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Theorem prsspwg 4173
Description: An unordered pair belongs to the power class of a class iff each member belongs to the class. (Contributed by Thierry Arnoux, 3-Oct-2016.) (Revised by NM, 18-Jan-2018.)
Assertion
Ref Expression
prsspwg  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( { A ,  B }  C_  ~P C  <->  ( A  C_  C  /\  B  C_  C ) ) )

Proof of Theorem prsspwg
StepHypRef Expression
1 prssg 4171 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ( A  e. 
~P C  /\  B  e.  ~P C )  <->  { A ,  B }  C_  ~P C ) )
2 elpwg 4007 . . 3  |-  ( A  e.  V  ->  ( A  e.  ~P C  <->  A 
C_  C ) )
3 elpwg 4007 . . 3  |-  ( B  e.  W  ->  ( B  e.  ~P C  <->  B 
C_  C ) )
42, 3bi2anan9 871 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ( A  e. 
~P C  /\  B  e.  ~P C )  <->  ( A  C_  C  /\  B  C_  C ) ) )
51, 4bitr3d 255 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( { A ,  B }  C_  ~P C  <->  ( A  C_  C  /\  B  C_  C ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    e. wcel 1823    C_ wss 3461   ~Pcpw 3999   {cpr 4018
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-v 3108  df-un 3466  df-in 3468  df-ss 3475  df-pw 4001  df-sn 4017  df-pr 4019
This theorem is referenced by:  prsspw  4188
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