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Theorem uniexr 7138
Description: Converse of the Axiom of Union. Note that it does not require ax-un 7115. (Contributed by NM, 11-Nov-2003.)
Assertion
Ref Expression
uniexr ( 𝐴𝑉𝐴 ∈ V)

Proof of Theorem uniexr
StepHypRef Expression
1 pwuni 4626 . 2 𝐴 ⊆ 𝒫 𝐴
2 pwexg 4999 . 2 ( 𝐴𝑉 → 𝒫 𝐴 ∈ V)
3 ssexg 4956 . 2 ((𝐴 ⊆ 𝒫 𝐴 ∧ 𝒫 𝐴 ∈ V) → 𝐴 ∈ V)
41, 2, 3sylancr 698 1 ( 𝐴𝑉𝐴 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2139  Vcvv 3340  wss 3715  𝒫 cpw 4302   cuni 4588
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-pow 4992
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-v 3342  df-in 3722  df-ss 3729  df-pw 4304  df-uni 4589
This theorem is referenced by:  uniexb  7139  ssonprc  7158  ac5num  9069  bj-restv  33372  bj-mooreset  33380
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