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Theorem prsspwg 3591
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 3589 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ( A  e. 
~P C  /\  B  e.  ~P C )  <->  { A ,  B }  C_  ~P C ) )
2 elpwg 3433 . . 3  |-  ( A  e.  V  ->  ( A  e.  ~P C  <->  A 
C_  C ) )
3 elpwg 3433 . . 3  |-  ( B  e.  W  ->  ( B  e.  ~P C  <->  B 
C_  C ) )
42, 3bi2anan9 573 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( ( A  e. 
~P C  /\  B  e.  ~P C )  <->  ( A  C_  C  /\  B  C_  C ) ) )
51, 4bitr3d 188 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 102    <-> wb 103    e. wcel 1438    C_ wss 2997   ~Pcpw 3425   {cpr 3442
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448
This theorem is referenced by: (None)
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