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Theorem pwexb 4474
Description: The Axiom of Power Sets and its converse. A class is a set iff its power class is a set. (Contributed by NM, 11-Nov-2003.)
Assertion
Ref Expression
pwexb (𝐴 ∈ V ↔ 𝒫 𝐴 ∈ V)

Proof of Theorem pwexb
StepHypRef Expression
1 uniexb 4473 . 2 (𝒫 𝐴 ∈ V ↔ 𝒫 𝐴 ∈ V)
2 unipw 4217 . . 3 𝒫 𝐴 = 𝐴
32eleq1i 2243 . 2 ( 𝒫 𝐴 ∈ V ↔ 𝐴 ∈ V)
41, 3bitr2i 185 1 (𝐴 ∈ V ↔ 𝒫 𝐴 ∈ V)
Colors of variables: wff set class
Syntax hints:  wb 105  wcel 2148  Vcvv 2737  𝒫 cpw 3575   cuni 3809
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-un 4433
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-v 2739  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-uni 3810
This theorem is referenced by: (None)
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