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Theorem uniexr 7702
Description: Converse of the Axiom of Union. Note that it does not require ax-un 7677. (Contributed by NM, 11-Nov-2003.)
Assertion
Ref Expression
uniexr ( 𝐴𝑉𝐴 ∈ V)

Proof of Theorem uniexr
StepHypRef Expression
1 pwuni 4911 . 2 𝐴 ⊆ 𝒫 𝐴
2 pwexg 5338 . 2 ( 𝐴𝑉 → 𝒫 𝐴 ∈ V)
3 ssexg 5285 . 2 ((𝐴 ⊆ 𝒫 𝐴 ∧ 𝒫 𝐴 ∈ V) → 𝐴 ∈ V)
41, 2, 3sylancr 588 1 ( 𝐴𝑉𝐴 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2107  Vcvv 3448  wss 3915  𝒫 cpw 4565   cuni 4870
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708  ax-sep 5261  ax-pow 5325
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-rab 3411  df-v 3450  df-in 3922  df-ss 3932  df-pw 4567  df-uni 4871
This theorem is referenced by:  uniexb  7703  ssonprc  7727  ac5num  9979  bj-restv  35595  bj-mooreset  35602  ipoglb0  47093
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