Theorem uniexr 6972

Description: Converse of the Axiom of Union. Note that it does not require ax-un 6949. (Contributed by NM, 11-Nov-2003.)

Assertion

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

Proof of Theorem uniexr

Step	Hyp	Ref	Expression
1		pwuni 4474	. 2 ⊢ 𝐴 ⊆ 𝒫 ∪ 𝐴
2		pwexg 4850	. 2 ⊢ (∪ 𝐴 ∈ 𝑉 → 𝒫 ∪ 𝐴 ∈ V)
3		ssexg 4804	. 2 ⊢ ((𝐴 ⊆ 𝒫 ∪ 𝐴 ∧ 𝒫 ∪ 𝐴 ∈ V) → 𝐴 ∈ V)
4	1, 2, 3	sylancr 695	1 ⊢ (∪ 𝐴 ∈ 𝑉 → 𝐴 ∈ V)

Syntax hints:   → wi 4   ∈ wcel 1990  Vcvv 3200   ⊆ wss 3574  𝒫 cpw 4158  ∪ cuni 4436

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-pow 4843

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-in 3581  df-ss 3588  df-pw 4160  df-uni 4437

This theorem is referenced by:  uniexb  6973  ssonprc  6992  ac5num  8859  bj-restv  33048  bj-mooreset  33056