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Theorem difabs 4221
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 4217 . 2 (𝐴 ∖ (𝐵𝐵)) = ((𝐴𝐵) ∖ 𝐵)
2 unidm 4082 . . 3 (𝐵𝐵) = 𝐵
32difeq2i 4050 . 2 (𝐴 ∖ (𝐵𝐵)) = (𝐴𝐵)
41, 3eqtr3i 2826 1 ((𝐴𝐵) ∖ 𝐵) = (𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1538  cdif 3881  cun 3882
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-ext 2773
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-rab 3118  df-v 3446  df-dif 3887  df-un 3889  df-in 3891
This theorem is referenced by:  axcclem  9872  lpdifsn  21751  compne  41132
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