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Theorem disjdif2 4389
 Description: The difference of a class and a class disjoint from it is the original class. (Contributed by BJ, 21-Apr-2019.)
Assertion
Ref Expression
disjdif2 ((𝐴𝐵) = ∅ → (𝐴𝐵) = 𝐴)

Proof of Theorem disjdif2
StepHypRef Expression
1 difeq2 4047 . 2 ((𝐴𝐵) = ∅ → (𝐴 ∖ (𝐴𝐵)) = (𝐴 ∖ ∅))
2 difin 4191 . 2 (𝐴 ∖ (𝐴𝐵)) = (𝐴𝐵)
3 dif0 4289 . 2 (𝐴 ∖ ∅) = 𝐴
41, 2, 33eqtr3g 2859 1 ((𝐴𝐵) = ∅ → (𝐴𝐵) = 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∖ cdif 3881   ∩ cin 3883  ∅c0 4246 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-ex 1782  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-rab 3118  df-v 3446  df-dif 3887  df-in 3891  df-ss 3901  df-nul 4247 This theorem is referenced by:  opwo0id  5355  setsfun0  16514  cnfldfunALT  20107  ptbasfi  22189  undif5  30294  fzdif2  30543  fzodif2  30544  chtvalz  32008  bj-2upln1upl  34455  gneispace  40824  dvmptfprodlem  42573
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