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Theorem difabs 4122
 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 4118 . 2 (𝐴 ∖ (𝐵𝐵)) = ((𝐴𝐵) ∖ 𝐵)
2 unidm 3984 . . 3 (𝐵𝐵) = 𝐵
32difeq2i 3953 . 2 (𝐴 ∖ (𝐵𝐵)) = (𝐴𝐵)
41, 3eqtr3i 2852 1 ((𝐴𝐵) ∖ 𝐵) = (𝐴𝐵)
 Colors of variables: wff setvar class Syntax hints:   = wceq 1658   ∖ cdif 3796   ∪ cun 3797 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-ext 2804 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ral 3123  df-rab 3127  df-v 3417  df-dif 3802  df-un 3804  df-in 3806 This theorem is referenced by:  axcclem  9595  lpdifsn  21319  compne  39483  compneOLD  39484
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