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Theorem prsspwg 3647
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 3645 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ( A  e. 
~P C  /\  B  e.  ~P C )  <->  { A ,  B }  C_  ~P C ) )
2 elpwg 3486 . . 3  |-  ( A  e.  V  ->  ( A  e.  ~P C  <->  A 
C_  C ) )
3 elpwg 3486 . . 3  |-  ( B  e.  W  ->  ( B  e.  ~P C  <->  B 
C_  C ) )
42, 3bi2anan9 578 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ( A  e. 
~P C  /\  B  e.  ~P C )  <->  ( A  C_  C  /\  B  C_  C ) ) )
51, 4bitr3d 189 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( { A ,  B }  C_  ~P C  <->  ( A  C_  C  /\  B  C_  C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    e. wcel 1463    C_ wss 3039   ~Pcpw 3478   {cpr 3496
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502
This theorem is referenced by: (None)
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