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Theorem nelss 3882
 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 3814 . . 3 (𝐵𝐶 → (𝐴𝐵𝐴𝐶))
21com12 32 . 2 (𝐴𝐵 → (𝐵𝐶𝐴𝐶))
32con3dimp 399 1 ((𝐴𝐵 ∧ ¬ 𝐴𝐶) → ¬ 𝐵𝐶)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 386   ∈ wcel 2106   ⊆ wss 3791 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-ext 2753 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2763  df-cleq 2769  df-clel 2773  df-in 3798  df-ss 3805 This theorem is referenced by:  nrelv  5471  ordtr3  6021  frlmssuvc2  20538  clsk1indlem1  39292  mapssbi  40319  fourierdlem10  41254  salgensscntex  41479
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