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

Theorem ssneldd 3158
Description: If an element is not in a class, it is also not in a subclass of that class. Deduction form. (Contributed by David Moews, 1-May-2017.)
Hypotheses
Ref Expression
ssneld.1 (𝜑𝐴𝐵)
ssneldd.2 (𝜑 → ¬ 𝐶𝐵)
Assertion
Ref Expression
ssneldd (𝜑 → ¬ 𝐶𝐴)

Proof of Theorem ssneldd
StepHypRef Expression
1 ssneldd.2 . 2 (𝜑 → ¬ 𝐶𝐵)
2 ssneld.1 . . 3 (𝜑𝐴𝐵)
32ssneld 3157 . 2 (𝜑 → (¬ 𝐶𝐵 → ¬ 𝐶𝐴))
41, 3mpd 13 1 (𝜑 → ¬ 𝐶𝐴)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wcel 2148  wss 3129
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-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  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-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-in 3135  df-ss 3142
This theorem is referenced by:  0nelrel  4668  addnqprlemfl  7536  addnqprlemfu  7537  mulnqprlemfl  7552  mulnqprlemfu  7553  cauappcvgprlemladdru  7633  fprodntrivap  11563  fprodssdc  11569
  Copyright terms: Public domain W3C validator