Theorem uniexb 7767

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 7746	. 2 ⊢ (𝐴 ∈ V → ∪ 𝐴 ∈ V)
2		uniexr 7766	. 2 ⊢ (∪ 𝐴 ∈ V → 𝐴 ∈ V)
3	1, 2	impbii 208	1 ⊢ (𝐴 ∈ V ↔ ∪ 𝐴 ∈ V)

Syntax hints:   ↔ wb 205   ∈ wcel 2098  Vcvv 3461  ∪ cuni 4909

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-sep 5300  ax-pow 5365  ax-un 7741

This theorem depends on definitions:  df-bi 206  df-an 395  df-3an 1086  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-rab 3419  df-v 3463  df-in 3951  df-ss 3961  df-pw 4606  df-uni 4910

This theorem is referenced by:  elpwpwel  7770  ixpexg  8941  rankuni  9888  unialeph  10126  ttukeylem1  10534  tgss2  22934  ordtbas2  23139  ordtbas  23140  ordttopon  23141  ordtopn1  23142  ordtopn2  23143  ordtrest2  23152  isref  23457  islocfin  23465  txbasex  23514  ptbasin2  23526  ordthmeolem  23749  alexsublem  23992  alexsub  23993  alexsubb  23994  ussid  24209  ordtrest2NEW  33652  brbigcup  35622  isfne  35951  isfne4  35952  isfne4b  35953  fnessref  35969  neibastop1  35971  fnejoin2  35981  prtex  38479