Theorem uniexb 7718

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 7694	. 2 ⊢ (𝐴 ∈ V → ∪ 𝐴 ∈ V)
2		uniexr 7717	. 2 ⊢ (∪ 𝐴 ∈ V → 𝐴 ∈ V)
3	1, 2	impbii 209	1 ⊢ (𝐴 ∈ V ↔ ∪ 𝐴 ∈ V)

Syntax hints:   ↔ wb 206   ∈ wcel 2114  Vcvv 3429  ∪ cuni 4850

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 2708  ax-sep 5231  ax-pow 5307  ax-un 7689

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 2715  df-cleq 2728  df-clel 2811  df-rab 3390  df-v 3431  df-in 3896  df-ss 3906  df-pw 4543  df-uni 4851

This theorem is referenced by:  elpwpwel  7721  ixpexg  8870  rankuni  9787  unialeph  10023  ttukeylem1  10431  tgss2  22952  ordtbas2  23156  ordtbas  23157  ordttopon  23158  ordtopn1  23159  ordtopn2  23160  ordtrest2  23169  isref  23474  islocfin  23482  txbasex  23531  ptbasin2  23543  ordthmeolem  23766  alexsublem  24009  alexsub  24010  alexsubb  24011  ussid  24225  ordtrest2NEW  34067  brbigcup  36078  isfne  36521  isfne4  36522  isfne4b  36523  fnessref  36539  neibastop1  36541  fnejoin2  36551  prtex  39326