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Theorem nssne1 3857
Description: Two classes are different if they don't include the same class. (Contributed by NM, 23-Apr-2015.)
Assertion
Ref Expression
nssne1 ((𝐴𝐵 ∧ ¬ 𝐴𝐶) → 𝐵𝐶)

Proof of Theorem nssne1
StepHypRef Expression
1 sseq2 3823 . . . 4 (𝐵 = 𝐶 → (𝐴𝐵𝐴𝐶))
21biimpcd 241 . . 3 (𝐴𝐵 → (𝐵 = 𝐶𝐴𝐶))
32necon3bd 2985 . 2 (𝐴𝐵 → (¬ 𝐴𝐶𝐵𝐶))
43imp 396 1 ((𝐴𝐵 ∧ ¬ 𝐴𝐶) → 𝐵𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 385   = wceq 1653  wne 2971  wss 3769
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-ext 2777
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-ne 2972  df-in 3776  df-ss 3783
This theorem is referenced by: (None)
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