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Theorem uniexr 7746
Description: Converse of the Axiom of Union. Note that it does not require ax-un 7721. (Contributed by NM, 11-Nov-2003.)
Assertion
Ref Expression
uniexr ( 𝐴𝑉𝐴 ∈ V)

Proof of Theorem uniexr
StepHypRef Expression
1 pwuni 4942 . 2 𝐴 ⊆ 𝒫 𝐴
2 pwexg 5369 . 2 ( 𝐴𝑉 → 𝒫 𝐴 ∈ V)
3 ssexg 5316 . 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 2098  Vcvv 3468  wss 3943  𝒫 cpw 4597   cuni 4902
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-sep 5292  ax-pow 5356
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-rab 3427  df-v 3470  df-in 3950  df-ss 3960  df-pw 4599  df-uni 4903
This theorem is referenced by:  uniexb  7747  ssonprc  7771  ac5num  10030  bj-restv  36483  bj-mooreset  36490  ipoglb0  47874
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