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Theorem pwexb 4406
 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 4405 . 2 (𝒫 𝐴 ∈ V ↔ 𝒫 𝐴 ∈ V)
2 unipw 4150 . . 3 𝒫 𝐴 = 𝐴
32eleq1i 2207 . 2 ( 𝒫 𝐴 ∈ V ↔ 𝐴 ∈ V)
41, 3bitr2i 184 1 (𝐴 ∈ V ↔ 𝒫 𝐴 ∈ V)
 Colors of variables: wff set class Syntax hints:   ↔ wb 104   ∈ wcel 2112  Vcvv 2691  𝒫 cpw 3517  ∪ cuni 3746 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-13 2114  ax-14 2115  ax-ext 2123  ax-sep 4056  ax-pow 4108  ax-un 4366 This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1732  df-clab 2128  df-cleq 2134  df-clel 2137  df-nfc 2272  df-rex 2424  df-v 2693  df-in 3084  df-ss 3091  df-pw 3519  df-sn 3540  df-uni 3747 This theorem is referenced by: (None)
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