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

Theorem unundir 3808
 Description: Union distributes over itself. (Contributed by NM, 17-Aug-2004.)
Assertion
Ref Expression
unundir ((𝐴𝐵) ∪ 𝐶) = ((𝐴𝐶) ∪ (𝐵𝐶))

Proof of Theorem unundir
StepHypRef Expression
1 unidm 3789 . . 3 (𝐶𝐶) = 𝐶
21uneq2i 3797 . 2 ((𝐴𝐵) ∪ (𝐶𝐶)) = ((𝐴𝐵) ∪ 𝐶)
3 un4 3806 . 2 ((𝐴𝐵) ∪ (𝐶𝐶)) = ((𝐴𝐶) ∪ (𝐵𝐶))
42, 3eqtr3i 2675 1 ((𝐴𝐵) ∪ 𝐶) = ((𝐴𝐶) ∪ (𝐵𝐶))
 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:  iocunico  38113
 Copyright terms: Public domain W3C validator