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Theorem nssne2 4012
 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 3976 . . . 4 (𝐴 = 𝐵 → (𝐴𝐶𝐵𝐶))
21biimpcd 252 . . 3 (𝐴𝐶 → (𝐴 = 𝐵𝐵𝐶))
32necon3bd 3027 . 2 (𝐴𝐶 → (¬ 𝐵𝐶𝐴𝐵))
43imp 410 1 ((𝐴𝐶 ∧ ¬ 𝐵𝐶) → 𝐴𝐵)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   = wceq 1538   ≠ wne 3013   ⊆ wss 3918 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-12 2179  ax-ext 2796 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-ne 3014  df-v 3481  df-in 3925  df-ss 3935 This theorem is referenced by:  atcvatlem  30157  mdsymlem3  30177  disjdifprg  30322  mapdh6aN  38931  mapdh8e  38980  hdmap1l6a  39005
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