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Theorem pwexb 4270
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 4269 . 2 (𝒫 𝐴 ∈ V ↔ 𝒫 𝐴 ∈ V)
2 unipw 4018 . . 3 𝒫 𝐴 = 𝐴
32eleq1i 2150 . 2 ( 𝒫 𝐴 ∈ V ↔ 𝐴 ∈ V)
41, 3bitr2i 183 1 (𝐴 ∈ V ↔ 𝒫 𝐴 ∈ V)
Colors of variables: wff set class
Syntax hints:  wb 103  wcel 1436  Vcvv 2615  𝒫 cpw 3415   cuni 3636
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 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3932  ax-pow 3984  ax-un 4234
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-rex 2361  df-v 2617  df-in 2994  df-ss 3001  df-pw 3417  df-sn 3437  df-uni 3637
This theorem is referenced by: (None)
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