Theorem uniexr 7613

Description: Converse of the Axiom of Union. Note that it does not require ax-un 7588. (Contributed by NM, 11-Nov-2003.)

Assertion

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

Proof of Theorem uniexr

Step	Hyp	Ref	Expression
1		pwuni 4878	. 2 ⊢ 𝐴 ⊆ 𝒫 ∪ 𝐴
2		pwexg 5301	. 2 ⊢ (∪ 𝐴 ∈ 𝑉 → 𝒫 ∪ 𝐴 ∈ V)
3		ssexg 5247	. 2 ⊢ ((𝐴 ⊆ 𝒫 ∪ 𝐴 ∧ 𝒫 ∪ 𝐴 ∈ V) → 𝐴 ∈ V)
4	1, 2, 3	sylancr 587	1 ⊢ (∪ 𝐴 ∈ 𝑉 → 𝐴 ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2106  Vcvv 3432   ⊆ wss 3887  𝒫 cpw 4533  ∪ cuni 4839

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5223  ax-pow 5288

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-rab 3073  df-v 3434  df-in 3894  df-ss 3904  df-pw 4535  df-uni 4840

This theorem is referenced by:  uniexb  7614  ssonprc  7637  ac5num  9792  bj-restv  35266  bj-mooreset  35273  ipoglb0  46280