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Theorem nelss 4014
 Description: Demonstrate by witnesses that two classes lack a subclass relation. (Contributed by Stefan O'Rear, 5-Feb-2015.)
Assertion
Ref Expression
nelss ((𝐴𝐵 ∧ ¬ 𝐴𝐶) → ¬ 𝐵𝐶)

Proof of Theorem nelss
StepHypRef Expression
1 ssel 3945 . . 3 (𝐵𝐶 → (𝐴𝐵𝐴𝐶))
21com12 32 . 2 (𝐴𝐵 → (𝐵𝐶𝐴𝐶))
32con3dimp 412 1 ((𝐴𝐵 ∧ ¬ 𝐴𝐶) → ¬ 𝐵𝐶)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∈ wcel 2115   ⊆ 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-v 3481  df-in 3925  df-ss 3935 This theorem is referenced by:  nrelv  5654  ordtr3  6217  smndex2dnrinv  18069  frlmssuvc2  20925  clsk1indlem1  40583  mapssbi  41683  fourierdlem10  42601  salgensscntex  42826
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