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Theorem prsspw 3527
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 Andrew Salmon, 26-Jun-2011.)
Hypotheses
Ref Expression
prsspw.1  _V
prsspw.2  _V
Assertion
Ref Expression
prsspw  { ,  }  C_ 
~P C 
C_  C  C_  C

Proof of Theorem prsspw
StepHypRef Expression
1 prsspw.1 . . 3  _V
2 prsspw.2 . . 3  _V
31, 2prss 3511 . 2  ~P C  ~P C  { ,  }  C_  ~P C
41elpw 3357 . . 3  ~P C  C_  C
52elpw 3357 . . 3  ~P C  C_  C
64, 5anbi12i 433 . 2  ~P C  ~P C  C_  C  C_  C
73, 6bitr3i 175 1  { ,  }  C_ 
~P C 
C_  C  C_  C
Colors of variables: wff set class
Syntax hints:   wa 97   wb 98   wcel 1390   _Vcvv 2551    C_ wss 2911   ~Pcpw 3351   {cpr 3368
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374
This theorem is referenced by: (None)
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