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Theorem uniexr 7765
Description: Converse of the Axiom of Union. Note that it does not require ax-un 7740. (Contributed by NM, 11-Nov-2003.)
Assertion
Ref Expression
uniexr ( 𝐴𝑉𝐴 ∈ V)

Proof of Theorem uniexr
StepHypRef Expression
1 pwuni 4948 . 2 𝐴 ⊆ 𝒫 𝐴
2 pwexg 5378 . 2 ( 𝐴𝑉 → 𝒫 𝐴 ∈ V)
3 ssexg 5323 . 2 ((𝐴 ⊆ 𝒫 𝐴 ∧ 𝒫 𝐴 ∈ V) → 𝐴 ∈ V)
41, 2, 3sylancr 586 1 ( 𝐴𝑉𝐴 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2099  Vcvv 3471  wss 3947  𝒫 cpw 4603   cuni 4908
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-sep 5299  ax-pow 5365
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-rab 3430  df-v 3473  df-in 3954  df-ss 3964  df-pw 4605  df-uni 4909
This theorem is referenced by:  uniexb  7766  ssonprc  7790  ac5num  10060  bj-restv  36574  bj-mooreset  36581  ipoglb0  48005
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