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Theorem eldifd 3085
Description: If a class is in one class and not another, it is also in their difference. One-way deduction form of eldif 3084. (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 3084 . 2 (𝐴 ∈ (𝐵𝐶) ↔ (𝐴𝐵 ∧ ¬ 𝐴𝐶))
41, 2, 3sylanbrc 414 1 (𝜑𝐴 ∈ (𝐵𝐶))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wcel 1481  cdif 3072
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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-dif 3077
This theorem is referenced by:  exmidundif  4136  exmidundifim  4137  frirrg  4279  dcdifsnid  6407  phpelm  6767  findcard2d  6792  findcard2sd  6793  diffifi  6795  unsnfidcex  6815  unsnfidcel  6816  undifdcss  6818  difinfsnlem  6991  difinfsn  6992  hashunlem  10581  seq3coll  10616  fsum3cvg  11178  isumss  11191  fisumss  11192  fproddccvg  11372  sqrt2irr0  11876  logbgcd1irr  13090
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