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Theorem difabs 4035
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 4030 . 2 (𝐴 ∖ (𝐵𝐵)) = ((𝐴𝐵) ∖ 𝐵)
2 unidm 3899 . . 3 (𝐵𝐵) = 𝐵
32difeq2i 3868 . 2 (𝐴 ∖ (𝐵𝐵)) = (𝐴𝐵)
41, 3eqtr3i 2784 1 ((𝐴𝐵) ∖ 𝐵) = (𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1632  cdif 3712  cun 3713
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722
This theorem is referenced by:  axcclem  9471  lpdifsn  21149  compne  39145  compneOLD  39146
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