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

Theorem nssne2 2996
 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 2960 . . . 4 (A = B → (A𝐶B𝐶))
21biimpcd 148 . . 3 (A𝐶 → (A = BB𝐶))
32necon3bd 2242 . 2 (A𝐶 → (¬ B𝐶AB))
43imp 115 1 ((A𝐶 ¬ B𝐶) → AB)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   = wceq 1242   ≠ wne 2201   ⊆ wss 2911 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-ne 2203  df-in 2918  df-ss 2925 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator