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

Theorem uniun 4935
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 1880 . . . 4 (∃𝑦((𝑥𝑦𝑦𝐴) ∨ (𝑥𝑦𝑦𝐵)) ↔ (∃𝑦(𝑥𝑦𝑦𝐴) ∨ ∃𝑦(𝑥𝑦𝑦𝐵)))
2 elun 4163 . . . . . . 7 (𝑦 ∈ (𝐴𝐵) ↔ (𝑦𝐴𝑦𝐵))
32anbi2i 623 . . . . . 6 ((𝑥𝑦𝑦 ∈ (𝐴𝐵)) ↔ (𝑥𝑦 ∧ (𝑦𝐴𝑦𝐵)))
4 andi 1009 . . . . . 6 ((𝑥𝑦 ∧ (𝑦𝐴𝑦𝐵)) ↔ ((𝑥𝑦𝑦𝐴) ∨ (𝑥𝑦𝑦𝐵)))
53, 4bitri 275 . . . . 5 ((𝑥𝑦𝑦 ∈ (𝐴𝐵)) ↔ ((𝑥𝑦𝑦𝐴) ∨ (𝑥𝑦𝑦𝐵)))
65exbii 1845 . . . 4 (∃𝑦(𝑥𝑦𝑦 ∈ (𝐴𝐵)) ↔ ∃𝑦((𝑥𝑦𝑦𝐴) ∨ (𝑥𝑦𝑦𝐵)))
7 eluni 4915 . . . . 5 (𝑥 𝐴 ↔ ∃𝑦(𝑥𝑦𝑦𝐴))
8 eluni 4915 . . . . 5 (𝑥 𝐵 ↔ ∃𝑦(𝑥𝑦𝑦𝐵))
97, 8orbi12i 914 . . . 4 ((𝑥 𝐴𝑥 𝐵) ↔ (∃𝑦(𝑥𝑦𝑦𝐴) ∨ ∃𝑦(𝑥𝑦𝑦𝐵)))
101, 6, 93bitr4i 303 . . 3 (∃𝑦(𝑥𝑦𝑦 ∈ (𝐴𝐵)) ↔ (𝑥 𝐴𝑥 𝐵))
11 eluni 4915 . . 3 (𝑥 (𝐴𝐵) ↔ ∃𝑦(𝑥𝑦𝑦 ∈ (𝐴𝐵)))
12 elun 4163 . . 3 (𝑥 ∈ ( 𝐴 𝐵) ↔ (𝑥 𝐴𝑥 𝐵))
1310, 11, 123bitr4i 303 . 2 (𝑥 (𝐴𝐵) ↔ 𝑥 ∈ ( 𝐴 𝐵))
1413eqriv 2732 1 (𝐴𝐵) = ( 𝐴 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wa 395  wo 847   = wceq 1537  wex 1776  wcel 2106  cun 3961   cuni 4912
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-un 3968  df-uni 4913
This theorem is referenced by:  unidif0  5366  unisucs  6463  fvssunirnOLD  6941  fvun  6999  onuninsuci  7861  tc2  9780  fin1a2lem10  10447  fin1a2lem12  10449  incexclem  15869  dprd2da  20077  dmdprdsplit2lem  20080  ordtuni  23214  cmpcld  23426  uncmp  23427  refun0  23539  lfinun  23549  1stckgenlem  23577  filconn  23907  ufildr  23955  alexsubALTlem3  24073  cldsubg  24135  icccmplem2  24859  uniioombllem3  25634  madeoldsuc  27938  zs12bday  28439  sxbrsigalem0  34253  fiunelcarsg  34298  carsgclctunlem1  34299  carsggect  34300  cvmscld  35258  refssfne  36341  topjoin  36348  pibt2  37400  mbfresfi  37653  onsucunitp  43363  oaun3  43372  fourierdlem80  46142  isomenndlem  46486
  Copyright terms: Public domain W3C validator