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Theorem difabs 3201
 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 3197 . 2 (𝐴 ∖ (𝐵𝐵)) = ((𝐴𝐵) ∖ 𝐵)
2 unidm 3086 . . 3 (𝐵𝐵) = 𝐵
32difeq2i 3059 . 2 (𝐴 ∖ (𝐵𝐵)) = (𝐴𝐵)
41, 3eqtr3i 2062 1 ((𝐴𝐵) ∖ 𝐵) = (𝐴𝐵)
 Colors of variables: wff set class Syntax hints:   = wceq 1243   ∖ cdif 2914   ∪ cun 2915 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rab 2315  df-v 2559  df-dif 2920  df-un 2922  df-in 2924 This theorem is referenced by: (None)
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