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Theorem disjdif2 4191
 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 3865 . 2 ((𝐴𝐵) = ∅ → (𝐴 ∖ (𝐴𝐵)) = (𝐴 ∖ ∅))
2 difin 4004 . 2 (𝐴 ∖ (𝐴𝐵)) = (𝐴𝐵)
3 dif0 4093 . 2 (𝐴 ∖ ∅) = 𝐴
41, 2, 33eqtr3g 2817 1 ((𝐴𝐵) = ∅ → (𝐴𝐵) = 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1632   ∖ cdif 3712   ∩ cin 3714  ∅c0 4058 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rab 3059  df-v 3342  df-dif 3718  df-in 3722  df-ss 3729  df-nul 4059 This theorem is referenced by:  opwo0id  5109  setsfun0  16096  cnfldfunALT  19961  ptbasfi  21586  fzdif2  29860  fzodif2  29861  chtvalz  31016  bj-2upln1upl  33318  gneispace  38934  dvmptfprodlem  40662
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