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Theorem pwexb 4234
 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 4233 . 2 (𝒫 𝐴 ∈ V ↔ 𝒫 𝐴 ∈ V)
2 unipw 3981 . . 3 𝒫 𝐴 = 𝐴
32eleq1i 2119 . 2 ( 𝒫 𝐴 ∈ V ↔ 𝐴 ∈ V)
41, 3bitr2i 178 1 (𝐴 ∈ V ↔ 𝒫 𝐴 ∈ V)
 Colors of variables: wff set class Syntax hints:   ↔ wb 102   ∈ wcel 1409  Vcvv 2574  𝒫 cpw 3387  ∪ cuni 3608 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-un 4198 This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-rex 2329  df-v 2576  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-uni 3609 This theorem is referenced by: (None)
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