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Theorem eldifd 3139
Description: If a class is in one class and not another, it is also in their difference. One-way deduction form of eldif 3138. (Contributed by David Moews, 1-May-2017.)
Hypotheses
Ref Expression
eldifd.1 (𝜑𝐴𝐵)
eldifd.2 (𝜑 → ¬ 𝐴𝐶)
Assertion
Ref Expression
eldifd (𝜑𝐴 ∈ (𝐵𝐶))

Proof of Theorem eldifd
StepHypRef Expression
1 eldifd.1 . 2 (𝜑𝐴𝐵)
2 eldifd.2 . 2 (𝜑 → ¬ 𝐴𝐶)
3 eldif 3138 . 2 (𝐴 ∈ (𝐵𝐶) ↔ (𝐴𝐵 ∧ ¬ 𝐴𝐶))
41, 2, 3sylanbrc 417 1 (𝜑𝐴 ∈ (𝐵𝐶))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wcel 2148  cdif 3126
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-dif 3131
This theorem is referenced by:  exmidundif  4203  exmidundifim  4204  frirrg  4346  dcdifsnid  6498  phpelm  6859  findcard2d  6884  findcard2sd  6885  diffifi  6887  unsnfidcex  6912  unsnfidcel  6913  undifdcss  6915  difinfsnlem  7091  difinfsn  7092  hashunlem  10755  seq3coll  10793  fsum3cvg  11357  isumss  11370  fisumss  11371  fproddccvg  11551  fprodssdc  11569  sqrt2irr0  12134  nnoddn2prmb  12232  logbgcd1irr  14018
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