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

Theorem ssneldd 3068
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 3067 . 2 (𝜑 → (¬ 𝐶𝐵 → ¬ 𝐶𝐴))
41, 3mpd 13 1 (𝜑 → ¬ 𝐶𝐴)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wcel 1463  wss 3039
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 586  ax-in2 587  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-11 1467  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-in 3045  df-ss 3052
This theorem is referenced by:  0nelrel  4553  addnqprlemfl  7331  addnqprlemfu  7332  mulnqprlemfl  7347  mulnqprlemfu  7348  cauappcvgprlemladdru  7428
  Copyright terms: Public domain W3C validator