Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  eldifbd GIF version

Theorem eldifbd 2994
 Description: If a class is in the difference of two classes, it is not in the subtrahend. One-way deduction form of eldif 2991. (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 2991 . . 3 (𝐴 ∈ (𝐵𝐶) ↔ (𝐴𝐵 ∧ ¬ 𝐴𝐶))
31, 2sylib 120 . 2 (𝜑 → (𝐴𝐵 ∧ ¬ 𝐴𝐶))
43simprd 112 1 (𝜑 → ¬ 𝐴𝐶)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 102   ∈ wcel 1434   ∖ cdif 2979 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065 This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2612  df-dif 2984 This theorem is referenced by:  fidifsnen  6426  fiunsnnn  6437  unfidisj  6466  ssfirab  6475  fnfi  6478  hashunlem  9880  hashxp  9902
 Copyright terms: Public domain W3C validator