Theorem uniexr 7718

Description: Converse of the Axiom of Union. Note that it does not require ax-un 7690. (Contributed by NM, 11-Nov-2003.)

Assertion

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

Proof of Theorem uniexr

Step	Hyp	Ref	Expression
1		pwuni 4903	. 2 ⊢ 𝐴 ⊆ 𝒫 ∪ 𝐴
2		pwexg 5325	. 2 ⊢ (∪ 𝐴 ∈ 𝑉 → 𝒫 ∪ 𝐴 ∈ V)
3		ssexg 5270	. 2 ⊢ ((𝐴 ⊆ 𝒫 ∪ 𝐴 ∧ 𝒫 ∪ 𝐴 ∈ V) → 𝐴 ∈ V)
4	1, 2, 3	sylancr 588	1 ⊢ (∪ 𝐴 ∈ 𝑉 → 𝐴 ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2114  Vcvv 3442   ⊆ wss 3903  𝒫 cpw 4556  ∪ cuni 4865

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5243  ax-pow 5312

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1089  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3402  df-v 3444  df-in 3910  df-ss 3920  df-pw 4558  df-uni 4866

This theorem is referenced by:  uniexb  7719  ssonprc  7742  ac5num  9958  bj-restv  37348  bj-mooreset  37355  ipoglb0  49353