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Theorem uniexr 7465
 Description: Converse of the Axiom of Union. Note that it does not require ax-un 7441. (Contributed by NM, 11-Nov-2003.)
Assertion
Ref Expression
uniexr ( 𝐴𝑉𝐴 ∈ V)

Proof of Theorem uniexr
StepHypRef Expression
1 pwuni 4837 . 2 𝐴 ⊆ 𝒫 𝐴
2 pwexg 5244 . 2 ( 𝐴𝑉 → 𝒫 𝐴 ∈ V)
3 ssexg 5191 . 2 ((𝐴 ⊆ 𝒫 𝐴 ∧ 𝒫 𝐴 ∈ V) → 𝐴 ∈ V)
41, 2, 3sylancr 590 1 ( 𝐴𝑉𝐴 ∈ V)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2111  Vcvv 3441   ⊆ wss 3881  𝒫 cpw 4497  ∪ cuni 4800 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-pow 5231 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-rab 3115  df-v 3443  df-in 3888  df-ss 3898  df-pw 4499  df-uni 4801 This theorem is referenced by:  uniexb  7466  ssonprc  7487  ac5num  9447  bj-restv  34507  bj-mooreset  34514
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