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

Theorem undifabs 4409
Description: Absorption of difference by union. (Contributed by NM, 18-Aug-2013.)
Assertion
Ref Expression
undifabs (𝐴 ∪ (𝐴𝐵)) = 𝐴

Proof of Theorem undifabs
StepHypRef Expression
1 undif3 4250 . 2 (𝐴 ∪ (𝐴𝐵)) = ((𝐴𝐴) ∖ (𝐵𝐴))
2 unidm 4114 . . 3 (𝐴𝐴) = 𝐴
32difeq1i 4081 . 2 ((𝐴𝐴) ∖ (𝐵𝐴)) = (𝐴 ∖ (𝐵𝐴))
4 difdif 4093 . 2 (𝐴 ∖ (𝐵𝐴)) = 𝐴
51, 3, 43eqtri 2851 1 (𝐴 ∪ (𝐴𝐵)) = 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1538  cdif 3916  cun 3917
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-rab 3142  df-v 3482  df-dif 3922  df-un 3924
This theorem is referenced by:  dfif5  4466  indifundif  30296
  Copyright terms: Public domain W3C validator