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Theorem disjdif2 4419
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 4057 . 2 ((𝐴𝐵) = ∅ → (𝐴 ∖ (𝐴𝐵)) = (𝐴 ∖ ∅))
2 difin 4201 . 2 (𝐴 ∖ (𝐴𝐵)) = (𝐴𝐵)
3 dif0 4312 . 2 (𝐴 ∖ ∅) = 𝐴
41, 2, 33eqtr3g 2799 1 ((𝐴𝐵) = ∅ → (𝐴𝐵) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  cdif 3889  cin 3891  c0 4262
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2714  df-cleq 2728  df-clel 2814  df-rab 3287  df-v 3439  df-dif 3895  df-in 3899  df-nul 4263
This theorem is referenced by:  opwo0id  5424  setsfun0  16918  cnfldfun  20654  ptbasfi  22777  undif5  30912  fzdif2  31157  fzodif2  31158  chtvalz  32654  sltlpss  34132  bj-2upln1upl  35258  dvrelog2  40114  dvrelog3  40115  gneispace  41782  dvmptfprodlem  43534
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