Theorem uniexb 7747

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 7723	. 2 ⊢ (𝐴 ∈ V → ∪ 𝐴 ∈ V)
2		uniexr 7746	. 2 ⊢ (∪ 𝐴 ∈ V → 𝐴 ∈ V)
3	1, 2	impbii 211	1 ⊢ (𝐴 ∈ V ↔ ∪ 𝐴 ∈ V)

Syntax hints:   ↔ wb 208   ∈ wcel 2142  Vcvv 3454  ∪ cuni 4865

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5246  ax-pow 5322  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-3an 1100  df-tru 1563  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3415  df-v 3456  df-in 3911  df-ss 3921  df-pw 4557  df-uni 4866

This theorem is referenced by:  elpwpwel  7750  ixpexg  8904  rankuni  9821  unialeph  10057  ttukeylem1  10466  tgss2  23047  ordtbas2  23251  ordtbas  23252  ordttopon  23253  ordtopn1  23254  ordtopn2  23255  ordtrest2  23264  isref  23569  islocfin  23577  txbasex  23626  ptbasin2  23638  ordthmeolem  23861  alexsublem  24104  alexsub  24105  alexsubb  24106  ussid  24320  ordtrest2NEW  34220  brbigcup  36246  isfne  36699  isfne4  36700  isfne4b  36701  fnessref  36717  neibastop1  36719  fnejoin2  36729  prtex  39504