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Theorem prsspw 3565
 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
prsspw.2
Assertion
Ref Expression
prsspw

Proof of Theorem prsspw
StepHypRef Expression
1 prsspw.1 . . 3
2 prsspw.2 . . 3
31, 2prss 3549 . 2
41elpw 3396 . . 3
52elpw 3396 . . 3
64, 5anbi12i 448 . 2
73, 6bitr3i 184 1
 Colors of variables: wff set class Syntax hints:   wa 102   wb 103   wcel 1434  cvv 2602   wss 2974  cpw 3390  cpr 3407 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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064 This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413 This theorem is referenced by: (None)
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