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Theorem uniun 4392
 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 1799 . . . 4 (∃𝑦((𝑥𝑦𝑦𝐴) ∨ (𝑥𝑦𝑦𝐵)) ↔ (∃𝑦(𝑥𝑦𝑦𝐴) ∨ ∃𝑦(𝑥𝑦𝑦𝐵)))
2 elun 3715 . . . . . . 7 (𝑦 ∈ (𝐴𝐵) ↔ (𝑦𝐴𝑦𝐵))
32anbi2i 726 . . . . . 6 ((𝑥𝑦𝑦 ∈ (𝐴𝐵)) ↔ (𝑥𝑦 ∧ (𝑦𝐴𝑦𝐵)))
4 andi 907 . . . . . 6 ((𝑥𝑦 ∧ (𝑦𝐴𝑦𝐵)) ↔ ((𝑥𝑦𝑦𝐴) ∨ (𝑥𝑦𝑦𝐵)))
53, 4bitri 263 . . . . 5 ((𝑥𝑦𝑦 ∈ (𝐴𝐵)) ↔ ((𝑥𝑦𝑦𝐴) ∨ (𝑥𝑦𝑦𝐵)))
65exbii 1764 . . . 4 (∃𝑦(𝑥𝑦𝑦 ∈ (𝐴𝐵)) ↔ ∃𝑦((𝑥𝑦𝑦𝐴) ∨ (𝑥𝑦𝑦𝐵)))
7 eluni 4375 . . . . 5 (𝑥 𝐴 ↔ ∃𝑦(𝑥𝑦𝑦𝐴))
8 eluni 4375 . . . . 5 (𝑥 𝐵 ↔ ∃𝑦(𝑥𝑦𝑦𝐵))
97, 8orbi12i 542 . . . 4 ((𝑥 𝐴𝑥 𝐵) ↔ (∃𝑦(𝑥𝑦𝑦𝐴) ∨ ∃𝑦(𝑥𝑦𝑦𝐵)))
101, 6, 93bitr4i 291 . . 3 (∃𝑦(𝑥𝑦𝑦 ∈ (𝐴𝐵)) ↔ (𝑥 𝐴𝑥 𝐵))
11 eluni 4375 . . 3 (𝑥 (𝐴𝐵) ↔ ∃𝑦(𝑥𝑦𝑦 ∈ (𝐴𝐵)))
12 elun 3715 . . 3 (𝑥 ∈ ( 𝐴 𝐵) ↔ (𝑥 𝐴𝑥 𝐵))
1310, 11, 123bitr4i 291 . 2 (𝑥 (𝐴𝐵) ↔ 𝑥 ∈ ( 𝐴 𝐵))
1413eqriv 2607 1 (𝐴𝐵) = ( 𝐴 𝐵)
 Colors of variables: wff setvar class Syntax hints:   ∨ wo 382   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977   ∪ cun 3538  ∪ cuni 4372 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-un 3545  df-uni 4373 This theorem is referenced by:  unidif0  4764  unisuc  5718  fvssunirn  6127  fvun  6178  onuninsuci  6932  tc2  8501  fin1a2lem10  9114  fin1a2lem12  9116  incexclem  14407  dprd2da  18264  dmdprdsplit2lem  18267  ordtuni  20804  cmpcld  21015  uncmp  21016  refun0  21128  lfinun  21138  1stckgenlem  21166  filcon  21497  ufildr  21545  alexsubALTlem3  21663  cldsubg  21724  icccmplem2  22434  uniioombllem3  23159  sxbrsigalem0  29660  fiunelcarsg  29705  carsgclctunlem1  29706  carsggect  29707  cvmscld  30509  refssfne  31523  topjoin  31530  mbfresfi  32626  fourierdlem80  39079  isomenndlem  39420
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