MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  un23 Structured version   Visualization version   GIF version

Theorem un23 4184
Description: A rearrangement of union. (Contributed by NM, 12-Aug-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
un23 ((𝐴𝐵) ∪ 𝐶) = ((𝐴𝐶) ∪ 𝐵)

Proof of Theorem un23
StepHypRef Expression
1 unass 4182 . 2 ((𝐴𝐵) ∪ 𝐶) = (𝐴 ∪ (𝐵𝐶))
2 un12 4183 . 2 (𝐴 ∪ (𝐵𝐶)) = (𝐵 ∪ (𝐴𝐶))
3 uncom 4168 . 2 (𝐵 ∪ (𝐴𝐶)) = ((𝐴𝐶) ∪ 𝐵)
41, 2, 33eqtri 2767 1 ((𝐴𝐵) ∪ 𝐶) = ((𝐴𝐶) ∪ 𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  cun 3961
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
This theorem is referenced by:  ssunpr  4839  setscom  17214  cycpmco2rn  33128  poimirlem6  37613  poimirlem7  37614  poimirlem16  37623  poimirlem19  37626  iocunico  43200  dfrcl2  43664
  Copyright terms: Public domain W3C validator