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Theorem uniexr 7342
Description: Converse of the Axiom of Union. Note that it does not require ax-un 7319. (Contributed by NM, 11-Nov-2003.)
Assertion
Ref Expression
uniexr ( 𝐴𝑉𝐴 ∈ V)

Proof of Theorem uniexr
StepHypRef Expression
1 pwuni 4781 . 2 𝐴 ⊆ 𝒫 𝐴
2 pwexg 5170 . 2 ( 𝐴𝑉 → 𝒫 𝐴 ∈ V)
3 ssexg 5118 . 2 ((𝐴 ⊆ 𝒫 𝐴 ∧ 𝒫 𝐴 ∈ V) → 𝐴 ∈ V)
41, 2, 3sylancr 587 1 ( 𝐴𝑉𝐴 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2081  Vcvv 3437  wss 3859  𝒫 cpw 4453   cuni 4745
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-ext 2769  ax-sep 5094  ax-pow 5157
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-v 3439  df-in 3866  df-ss 3874  df-pw 4455  df-uni 4746
This theorem is referenced by:  uniexb  7343  ssonprc  7363  ac5num  9308  bj-restv  33985  bj-mooreset  33993
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