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

Theorem un23 3805
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 3803 . 2 ((𝐴𝐵) ∪ 𝐶) = (𝐴 ∪ (𝐵𝐶))
2 un12 3804 . 2 (𝐴 ∪ (𝐵𝐶)) = (𝐵 ∪ (𝐴𝐶))
3 uncom 3790 . 2 (𝐵 ∪ (𝐴𝐶)) = ((𝐴𝐶) ∪ 𝐵)
41, 2, 33eqtri 2677 1 ((𝐴𝐵) ∪ 𝐶) = ((𝐴𝐶) ∪ 𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1523  cun 3605
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-v 3233  df-un 3612
This theorem is referenced by:  ssunpr  4397  setscom  15950  poimirlem6  33545  poimirlem7  33546  poimirlem16  33555  poimirlem19  33558  iocunico  38113  dfrcl2  38283
  Copyright terms: Public domain W3C validator