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Theorem nssne1 4005
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 3971 . . . 4 (𝐵 = 𝐶 → (𝐴𝐵𝐴𝐶))
21biimpcd 249 . . 3 (𝐴𝐵 → (𝐵 = 𝐶𝐴𝐶))
32necon3bd 2954 . 2 (𝐴𝐵 → (¬ 𝐴𝐶𝐵𝐶))
43imp 408 1 ((𝐴𝐵 ∧ ¬ 𝐴𝐶) → 𝐵𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 397   = wceq 1542  wne 2940  wss 3911
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2941  df-v 3446  df-in 3918  df-ss 3928
This theorem is referenced by: (None)
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