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Theorem disjdif2 4479
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 4116 . 2 ((𝐴𝐵) = ∅ → (𝐴 ∖ (𝐴𝐵)) = (𝐴 ∖ ∅))
2 difin 4261 . 2 (𝐴 ∖ (𝐴𝐵)) = (𝐴𝐵)
3 dif0 4372 . 2 (𝐴 ∖ ∅) = 𝐴
41, 2, 33eqtr3g 2795 1 ((𝐴𝐵) = ∅ → (𝐴𝐵) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  cdif 3945  cin 3947  c0 4322
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-rab 3433  df-v 3476  df-dif 3951  df-in 3955  df-nul 4323
This theorem is referenced by:  undif5  4484  opwo0id  5497  setsfun0  17104  cnfldfun  20955  ptbasfi  23084  sltlpss  27398  fzdif2  31997  fzodif2  31998  chtvalz  33636  bj-2upln1upl  35900  disjresdif  37103  dvrelog2  40924  dvrelog3  40925  gneispace  42875  dvmptfprodlem  44650
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