Theorem uniexb 7712

Description: The Axiom of Union and its converse. A class is a set iff its union is a set. (Contributed by NM, 11-Nov-2003.)

Assertion

Ref	Expression
uniexb	⊢ (𝐴 ∈ V ↔ ∪ 𝐴 ∈ V)

Proof of Theorem uniexb

Step	Hyp	Ref	Expression
1		uniexg 7688	. 2 ⊢ (𝐴 ∈ V → ∪ 𝐴 ∈ V)
2		uniexr 7711	. 2 ⊢ (∪ 𝐴 ∈ V → 𝐴 ∈ V)
3	1, 2	impbii 209	1 ⊢ (𝐴 ∈ V ↔ ∪ 𝐴 ∈ V)

Syntax hints:   ↔ wb 206   ∈ wcel 2114  Vcvv 3430  ∪ cuni 4851

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 5232  ax-pow 5303  ax-un 7683

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 3391  df-v 3432  df-in 3897  df-ss 3907  df-pw 4544  df-uni 4852

This theorem is referenced by:  elpwpwel  7715  ixpexg  8864  rankuni  9781  unialeph  10017  ttukeylem1  10425  tgss2  22965  ordtbas2  23169  ordtbas  23170  ordttopon  23171  ordtopn1  23172  ordtopn2  23173  ordtrest2  23182  isref  23487  islocfin  23495  txbasex  23544  ptbasin2  23556  ordthmeolem  23779  alexsublem  24022  alexsub  24023  alexsubb  24024  ussid  24238  ordtrest2NEW  34086  brbigcup  36097  isfne  36540  isfne4  36541  isfne4b  36542  fnessref  36558  neibastop1  36560  fnejoin2  36570  prtex  39343