Theorem uniexr 7526

Description: Converse of the Axiom of Union. Note that it does not require ax-un 7501. (Contributed by NM, 11-Nov-2003.)

Assertion

Ref	Expression
uniexr	⊢ (∪ 𝐴 ∈ 𝑉 → 𝐴 ∈ V)

Proof of Theorem uniexr

Step	Hyp	Ref	Expression
1		pwuni 4844	. 2 ⊢ 𝐴 ⊆ 𝒫 ∪ 𝐴
2		pwexg 5256	. 2 ⊢ (∪ 𝐴 ∈ 𝑉 → 𝒫 ∪ 𝐴 ∈ V)
3		ssexg 5201	. 2 ⊢ ((𝐴 ⊆ 𝒫 ∪ 𝐴 ∧ 𝒫 ∪ 𝐴 ∈ V) → 𝐴 ∈ V)
4	1, 2, 3	sylancr 590	1 ⊢ (∪ 𝐴 ∈ 𝑉 → 𝐴 ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2112  Vcvv 3398   ⊆ wss 3853  𝒫 cpw 4499  ∪ cuni 4805

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-ext 2708  ax-sep 5177  ax-pow 5243

This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1546  df-ex 1788  df-sb 2073  df-clab 2715  df-cleq 2728  df-clel 2809  df-rab 3060  df-v 3400  df-in 3860  df-ss 3870  df-pw 4501  df-uni 4806

This theorem is referenced by:  uniexb  7527  ssonprc  7549  ac5num  9615  bj-restv  34950  bj-mooreset  34957  ipoglb0  45896