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

Theorem difabs 3925
Description: Absorption-like law for class difference: you can remove a class only once. (Contributed by FL, 2-Aug-2009.)
Assertion
Ref Expression
difabs ((𝐴𝐵) ∖ 𝐵) = (𝐴𝐵)

Proof of Theorem difabs
StepHypRef Expression
1 difun1 3920 . 2 (𝐴 ∖ (𝐵𝐵)) = ((𝐴𝐵) ∖ 𝐵)
2 unidm 3789 . . 3 (𝐵𝐵) = 𝐵
32difeq2i 3758 . 2 (𝐴 ∖ (𝐵𝐵)) = (𝐴𝐵)
41, 3eqtr3i 2675 1 ((𝐴𝐵) ∖ 𝐵) = (𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1523  cdif 3604  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-ral 2946  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614
This theorem is referenced by:  axcclem  9317  lpdifsn  20995  compne  38960  compneOLD  38961
  Copyright terms: Public domain W3C validator