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Theorem disjdif2 4342
 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 4014 . 2 ((𝐴𝐵) = ∅ → (𝐴 ∖ (𝐴𝐵)) = (𝐴 ∖ ∅))
2 difin 4158 . 2 (𝐴 ∖ (𝐴𝐵)) = (𝐴𝐵)
3 dif0 4252 . 2 (𝐴 ∖ ∅) = 𝐴
41, 2, 33eqtr3g 2854 1 ((𝐴𝐵) = ∅ → (𝐴𝐵) = 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1522   ∖ cdif 3856   ∩ cin 3858  ∅c0 4211 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-ext 2769 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ral 3110  df-rab 3114  df-v 3439  df-dif 3862  df-in 3866  df-ss 3874  df-nul 4212 This theorem is referenced by:  opwo0id  5278  setsfun0  16348  cnfldfunALT  20240  ptbasfi  21873  fzdif2  30200  fzodif2  30201  chtvalz  31517  bj-2upln1upl  33960  gneispace  39988  dvmptfprodlem  41790
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