ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  nssne2 GIF version

Theorem nssne2 3206
Description: Two classes are different if they are not subclasses of the same class. (Contributed by NM, 23-Apr-2015.)
Assertion
Ref Expression
nssne2 ((𝐴𝐶 ∧ ¬ 𝐵𝐶) → 𝐴𝐵)

Proof of Theorem nssne2
StepHypRef Expression
1 sseq1 3170 . . . 4 (𝐴 = 𝐵 → (𝐴𝐶𝐵𝐶))
21biimpcd 158 . . 3 (𝐴𝐶 → (𝐴 = 𝐵𝐵𝐶))
32necon3bd 2383 . 2 (𝐴𝐶 → (¬ 𝐵𝐶𝐴𝐵))
43imp 123 1 ((𝐴𝐶 ∧ ¬ 𝐵𝐶) → 𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103   = wceq 1348  wne 2340  wss 3121
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 609  ax-in2 610  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-11 1499  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-ne 2341  df-in 3127  df-ss 3134
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator