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Theorem eldifad 3024
Description: If a class is in the difference of two classes, it is also in the minuend. One-way deduction form of eldif 3022. (Contributed by David Moews, 1-May-2017.)
Hypothesis
Ref Expression
eldifad.1 (𝜑𝐴 ∈ (𝐵𝐶))
Assertion
Ref Expression
eldifad (𝜑𝐴𝐵)

Proof of Theorem eldifad
StepHypRef Expression
1 eldifad.1 . . 3 (𝜑𝐴 ∈ (𝐵𝐶))
2 eldif 3022 . . 3 (𝐴 ∈ (𝐵𝐶) ↔ (𝐴𝐵 ∧ ¬ 𝐴𝐶))
31, 2sylib 121 . 2 (𝜑 → (𝐴𝐵 ∧ ¬ 𝐴𝐶))
43simpld 111 1 (𝜑𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wcel 1445  cdif 3010
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077
This theorem depends on definitions:  df-bi 116  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-v 2635  df-dif 3015
This theorem is referenced by:  fimax2gtri  6697  finexdc  6698  unfidisj  6712  undifdc  6714  ssfirab  6723  fnfi  6726  iunfidisj  6735  hashunlem  10343  zfz1isolemiso  10375  fsumrelem  11030  iuncld  11983
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