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Theorem uniexr 7014
 Description: Converse of the Axiom of Union. Note that it does not require ax-un 6991. (Contributed by NM, 11-Nov-2003.)
Assertion
Ref Expression
uniexr ( 𝐴𝑉𝐴 ∈ V)

Proof of Theorem uniexr
StepHypRef Expression
1 pwuni 4506 . 2 𝐴 ⊆ 𝒫 𝐴
2 pwexg 4880 . 2 ( 𝐴𝑉 → 𝒫 𝐴 ∈ V)
3 ssexg 4837 . 2 ((𝐴 ⊆ 𝒫 𝐴 ∧ 𝒫 𝐴 ∈ V) → 𝐴 ∈ V)
41, 2, 3sylancr 696 1 ( 𝐴𝑉𝐴 ∈ V)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2030  Vcvv 3231   ⊆ wss 3607  𝒫 cpw 4191  ∪ cuni 4468 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-pow 4873 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-v 3233  df-in 3614  df-ss 3621  df-pw 4193  df-uni 4469 This theorem is referenced by:  uniexb  7015  ssonprc  7034  ac5num  8897  bj-restv  33173  bj-mooreset  33181
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