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Theorem nssne2 3859
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 3823 . . . 4 (𝐴 = 𝐵 → (𝐴𝐶𝐵𝐶))
21biimpcd 241 . . 3 (𝐴𝐶 → (𝐴 = 𝐵𝐵𝐶))
32necon3bd 2986 . 2 (𝐴𝐶 → (¬ 𝐵𝐶𝐴𝐵))
43imp 396 1 ((𝐴𝐶 ∧ ¬ 𝐵𝐶) → 𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 385   = wceq 1653  wne 2972  wss 3770
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 2778
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 2787  df-cleq 2793  df-clel 2796  df-ne 2973  df-in 3777  df-ss 3784
This theorem is referenced by:  atcvatlem  29768  mdsymlem3  29788  disjdifprg  29904  mapdh6aN  37755  mapdh8e  37804  hdmap1l6a  37829
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