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Theorem eldifbd 3053
Description: If a class is in the difference of two classes, it is not in the subtrahend. One-way deduction form of eldif 3050. (Contributed by David Moews, 1-May-2017.)
Hypothesis
Ref Expression
eldifbd.1 (𝜑𝐴 ∈ (𝐵𝐶))
Assertion
Ref Expression
eldifbd (𝜑 → ¬ 𝐴𝐶)

Proof of Theorem eldifbd
StepHypRef Expression
1 eldifbd.1 . . 3 (𝜑𝐴 ∈ (𝐵𝐶))
2 eldif 3050 . . 3 (𝐴 ∈ (𝐵𝐶) ↔ (𝐴𝐵 ∧ ¬ 𝐴𝐶))
31, 2sylib 121 . 2 (𝜑 → (𝐴𝐵 ∧ ¬ 𝐴𝐶))
43simprd 113 1 (𝜑 → ¬ 𝐴𝐶)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wcel 1465  cdif 3038
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-dif 3043
This theorem is referenced by:  fidifsnen  6732  fiunsnnn  6743  fimax2gtri  6763  unfidisj  6778  ssfirab  6790  fnfi  6793  iunfidisj  6802  hashunlem  10518  hashxp  10540  zfz1isolemiso  10550  fsumconst  11191  fsumrelem  11208  fsumcncntop  12652
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