Theorem uniexb 7751

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 7730	. 2 ⊢ (𝐴 ∈ V → ∪ 𝐴 ∈ V)
2		uniexr 7750	. 2 ⊢ (∪ 𝐴 ∈ V → 𝐴 ∈ V)
3	1, 2	impbii 208	1 ⊢ (𝐴 ∈ V ↔ ∪ 𝐴 ∈ V)

Syntax hints:   ↔ wb 205   ∈ wcel 2107  Vcvv 3475  ∪ cuni 4909

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-pow 5364  ax-un 7725

This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3434  df-v 3477  df-in 3956  df-ss 3966  df-pw 4605  df-uni 4910

This theorem is referenced by:  elpwpwel  7754  ixpexg  8916  rankuni  9858  unialeph  10096  ttukeylem1  10504  tgss2  22490  ordtbas2  22695  ordtbas  22696  ordttopon  22697  ordtopn1  22698  ordtopn2  22699  ordtrest2  22708  isref  23013  islocfin  23021  txbasex  23070  ptbasin2  23082  ordthmeolem  23305  alexsublem  23548  alexsub  23549  alexsubb  23550  ussid  23765  ordtrest2NEW  32934  brbigcup  34901  isfne  35272  isfne4  35273  isfne4b  35274  fnessref  35290  neibastop1  35292  fnejoin2  35302  prtex  37798