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Theorem uniexr 7754
Description: Converse of the Axiom of Union. Note that it does not require ax-un 7729. (Contributed by NM, 11-Nov-2003.)
Assertion
Ref Expression
uniexr ( 𝐴𝑉𝐴 ∈ V)

Proof of Theorem uniexr
StepHypRef Expression
1 pwuni 4949 . 2 𝐴 ⊆ 𝒫 𝐴
2 pwexg 5376 . 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 2105  Vcvv 3473  wss 3948  𝒫 cpw 4602   cuni 4908
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702  ax-sep 5299  ax-pow 5363
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-rab 3432  df-v 3475  df-in 3955  df-ss 3965  df-pw 4604  df-uni 4909
This theorem is referenced by:  uniexb  7755  ssonprc  7779  ac5num  10037  bj-restv  36443  bj-mooreset  36450  ipoglb0  47784
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