MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  uniexb Structured version   Visualization version   GIF version

Theorem uniexb 7703
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
StepHypRef Expression
1 uniexg 7682 . 2 (𝐴 ∈ V → 𝐴 ∈ V)
2 uniexr 7702 . 2 ( 𝐴 ∈ V → 𝐴 ∈ V)
31, 2impbii 208 1 (𝐴 ∈ V ↔ 𝐴 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wcel 2107  Vcvv 3448   cuni 4870
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 2708  ax-sep 5261  ax-pow 5325  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-rab 3411  df-v 3450  df-in 3922  df-ss 3932  df-pw 4567  df-uni 4871
This theorem is referenced by:  elpwpwel  7706  ixpexg  8867  rankuni  9806  unialeph  10044  ttukeylem1  10452  tgss2  22353  ordtbas2  22558  ordtbas  22559  ordttopon  22560  ordtopn1  22561  ordtopn2  22562  ordtrest2  22571  isref  22876  islocfin  22884  txbasex  22933  ptbasin2  22945  ordthmeolem  23168  alexsublem  23411  alexsub  23412  alexsubb  23413  ussid  23628  ordtrest2NEW  32544  brbigcup  34512  isfne  34840  isfne4  34841  isfne4b  34842  fnessref  34858  neibastop1  34860  fnejoin2  34870  prtex  37371
  Copyright terms: Public domain W3C validator