Theorem uniexb 7015

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 6997	. 2 ⊢ (𝐴 ∈ V → ∪ 𝐴 ∈ V)
2		uniexr 7014	. 2 ⊢ (∪ 𝐴 ∈ V → 𝐴 ∈ V)
3	1, 2	impbii 199	1 ⊢ (𝐴 ∈ V ↔ ∪ 𝐴 ∈ V)

Syntax hints:   ↔ wb 196   ∈ wcel 2030  Vcvv 3231  ∪ cuni 4468

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-pow 4873  ax-un 6991

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-rex 2947  df-v 3233  df-in 3614  df-ss 3621  df-pw 4193  df-uni 4469

This theorem is referenced by:  ixpexg  7974  rankuni  8764  unialeph  8962  ttukeylem1  9369  tgss2  20839  ordtbas2  21043  ordtbas  21044  ordttopon  21045  ordtopn1  21046  ordtopn2  21047  ordtrest2  21056  isref  21360  islocfin  21368  txbasex  21417  ptbasin2  21429  ordthmeolem  21652  alexsublem  21895  alexsub  21896  alexsubb  21897  ussid  22111  ordtrest2NEW  30097  omsfval  30484  brbigcup  32130  isfne  32459  isfne4  32460  isfne4b  32461  fnessref  32477  neibastop1  32479  fnejoin2  32489  prtex  34484