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Theorem prsspwg 4042
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 4040 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ( A  e. 
~P C  /\  B  e.  ~P C )  <->  { A ,  B }  C_  ~P C ) )
2 elpwg 3880 . . 3  |-  ( A  e.  V  ->  ( A  e.  ~P C  <->  A 
C_  C ) )
3 elpwg 3880 . . 3  |-  ( B  e.  W  ->  ( B  e.  ~P C  <->  B 
C_  C ) )
42, 3bi2anan9 868 . 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 369    e. wcel 1756    C_ wss 3340   ~Pcpw 3872   {cpr 3891
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 2423
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 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-v 2986  df-un 3345  df-in 3347  df-ss 3354  df-pw 3874  df-sn 3890  df-pr 3892
This theorem is referenced by: (None)
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