Theorem uniexr 7765

Description: Converse of the Axiom of Union. Note that it does not require ax-un 7740. (Contributed by NM, 11-Nov-2003.)

Assertion

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

Proof of Theorem uniexr

Step	Hyp	Ref	Expression
1		pwuni 4948	. 2 ⊢ 𝐴 ⊆ 𝒫 ∪ 𝐴
2		pwexg 5378	. 2 ⊢ (∪ 𝐴 ∈ 𝑉 → 𝒫 ∪ 𝐴 ∈ V)
3		ssexg 5323	. 2 ⊢ ((𝐴 ⊆ 𝒫 ∪ 𝐴 ∧ 𝒫 ∪ 𝐴 ∈ V) → 𝐴 ∈ V)
4	1, 2, 3	sylancr 586	1 ⊢ (∪ 𝐴 ∈ 𝑉 → 𝐴 ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2099  Vcvv 3471   ⊆ wss 3947  𝒫 cpw 4603  ∪ cuni 4908

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-sep 5299  ax-pow 5365

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-rab 3430  df-v 3473  df-in 3954  df-ss 3964  df-pw 4605  df-uni 4909

This theorem is referenced by:  uniexb  7766  ssonprc  7790  ac5num  10060  bj-restv  36574  bj-mooreset  36581  ipoglb0  48005