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Theorem disjdif2 3513
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 3259 . 2 ((𝐴𝐵) = ∅ → (𝐴 ∖ (𝐴𝐵)) = (𝐴 ∖ ∅))
2 difin 3384 . 2 (𝐴 ∖ (𝐴𝐵)) = (𝐴𝐵)
3 dif0 3505 . 2 (𝐴 ∖ ∅) = 𝐴
41, 2, 33eqtr3g 2243 1 ((𝐴𝐵) = ∅ → (𝐴𝐵) = 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1363  cdif 3138  cin 3140  c0 3434
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169
This theorem depends on definitions:  df-bi 117  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rab 2474  df-v 2751  df-dif 3143  df-in 3147  df-ss 3154  df-nul 3435
This theorem is referenced by:  setsfun0  12511
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