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Theorem disjdif2 4080
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 3755 . 2 ((𝐴𝐵) = ∅ → (𝐴 ∖ (𝐴𝐵)) = (𝐴 ∖ ∅))
2 difin 3894 . 2 (𝐴 ∖ (𝐴𝐵)) = (𝐴𝐵)
3 dif0 3983 . 2 (𝐴 ∖ ∅) = 𝐴
41, 2, 33eqtr3g 2708 1 ((𝐴𝐵) = ∅ → (𝐴𝐵) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1523  cdif 3604  cin 3606  c0 3948
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rab 2950  df-v 3233  df-dif 3610  df-in 3614  df-ss 3621  df-nul 3949
This theorem is referenced by:  opwo0id  4990  setsfun0  15941  cnfldfunALT  19807  ptbasfi  21432  fzdif2  29679  fzodif2  29680  chtvalz  30835  bj-2upln1upl  33137  gneispace  38749  dvmptfprodlem  40477
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