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Theorem uniun 4382
Description: The class union of the union of two classes. Theorem 8.3 of [Quine] p. 53. (Contributed by NM, 20-Aug-1993.)
Assertion
Ref Expression
uniun (𝐴𝐵) = ( 𝐴 𝐵)

Proof of Theorem uniun
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 19.43 1797 . . . 4 (∃𝑦((𝑥𝑦𝑦𝐴) ∨ (𝑥𝑦𝑦𝐵)) ↔ (∃𝑦(𝑥𝑦𝑦𝐴) ∨ ∃𝑦(𝑥𝑦𝑦𝐵)))
2 elun 3710 . . . . . . 7 (𝑦 ∈ (𝐴𝐵) ↔ (𝑦𝐴𝑦𝐵))
32anbi2i 725 . . . . . 6 ((𝑥𝑦𝑦 ∈ (𝐴𝐵)) ↔ (𝑥𝑦 ∧ (𝑦𝐴𝑦𝐵)))
4 andi 906 . . . . . 6 ((𝑥𝑦 ∧ (𝑦𝐴𝑦𝐵)) ↔ ((𝑥𝑦𝑦𝐴) ∨ (𝑥𝑦𝑦𝐵)))
53, 4bitri 262 . . . . 5 ((𝑥𝑦𝑦 ∈ (𝐴𝐵)) ↔ ((𝑥𝑦𝑦𝐴) ∨ (𝑥𝑦𝑦𝐵)))
65exbii 1762 . . . 4 (∃𝑦(𝑥𝑦𝑦 ∈ (𝐴𝐵)) ↔ ∃𝑦((𝑥𝑦𝑦𝐴) ∨ (𝑥𝑦𝑦𝐵)))
7 eluni 4365 . . . . 5 (𝑥 𝐴 ↔ ∃𝑦(𝑥𝑦𝑦𝐴))
8 eluni 4365 . . . . 5 (𝑥 𝐵 ↔ ∃𝑦(𝑥𝑦𝑦𝐵))
97, 8orbi12i 541 . . . 4 ((𝑥 𝐴𝑥 𝐵) ↔ (∃𝑦(𝑥𝑦𝑦𝐴) ∨ ∃𝑦(𝑥𝑦𝑦𝐵)))
101, 6, 93bitr4i 290 . . 3 (∃𝑦(𝑥𝑦𝑦 ∈ (𝐴𝐵)) ↔ (𝑥 𝐴𝑥 𝐵))
11 eluni 4365 . . 3 (𝑥 (𝐴𝐵) ↔ ∃𝑦(𝑥𝑦𝑦 ∈ (𝐴𝐵)))
12 elun 3710 . . 3 (𝑥 ∈ ( 𝐴 𝐵) ↔ (𝑥 𝐴𝑥 𝐵))
1310, 11, 123bitr4i 290 . 2 (𝑥 (𝐴𝐵) ↔ 𝑥 ∈ ( 𝐴 𝐵))
1413eqriv 2602 1 (𝐴𝐵) = ( 𝐴 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wo 381  wa 382   = wceq 1474  wex 1694  wcel 1975  cun 3533   cuni 4362
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-v 3170  df-un 3540  df-uni 4363
This theorem is referenced by:  unidif0  4755  unisuc  5700  fvssunirn  6108  fvun  6159  onuninsuci  6905  tc2  8474  fin1a2lem10  9087  fin1a2lem12  9089  incexclem  14349  dprd2da  18206  dmdprdsplit2lem  18209  ordtuni  20742  cmpcld  20953  uncmp  20954  refun0  21066  lfinun  21076  1stckgenlem  21104  filcon  21435  ufildr  21483  alexsubALTlem3  21601  cldsubg  21662  icccmplem2  22362  uniioombllem3  23072  sxbrsigalem0  29462  fiunelcarsg  29507  carsgclctunlem1  29508  carsggect  29509  cvmscld  30311  refssfne  31325  topjoin  31332  mbfresfi  32425  fourierdlem80  38879  isomenndlem  39220
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