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Theorem disjdif2 4480
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 4120 . 2 ((𝐴𝐵) = ∅ → (𝐴 ∖ (𝐴𝐵)) = (𝐴 ∖ ∅))
2 difin 4272 . 2 (𝐴 ∖ (𝐴𝐵)) = (𝐴𝐵)
3 dif0 4378 . 2 (𝐴 ∖ ∅) = 𝐴
41, 2, 33eqtr3g 2800 1 ((𝐴𝐵) = ∅ → (𝐴𝐵) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  cdif 3948  cin 3950  c0 4333
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-in 3958  df-nul 4334
This theorem is referenced by:  undif5  4485  opwo0id  5502  setsfun0  17209  cnfldfun  21378  cnfldfunOLD  21391  ptbasfi  23589  sltlpss  27945  slelss  27946  fzdif2  32792  fzodif2  32793  chtvalz  34644  bj-2upln1upl  37025  disjresdif  38243  dvrelog2  42065  dvrelog3  42066  readvrec2  42391  readvrec  42392  gneispace  44147  dvmptfprodlem  45959
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