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Theorem prsspw 4204
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 OpenAI, 25-Mar-2020.)
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 . 2  |-  A  e. 
_V
2 prsspw.2 . 2  |-  B  e. 
_V
3 prsspwg 4189 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( { A ,  B }  C_  ~P C  <->  ( A  C_  C  /\  B  C_  C ) ) )
41, 2, 3mp2an 672 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 1819   _Vcvv 3109    C_ wss 3471   ~Pcpw 4015   {cpr 4034
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-v 3111  df-un 3476  df-in 3478  df-ss 3485  df-pw 4017  df-sn 4033  df-pr 4035
This theorem is referenced by:  altxpsspw  29789
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