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Theorem nelss 4042
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 3970 . . 3 (𝐵𝐶 → (𝐴𝐵𝐴𝐶))
21com12 32 . 2 (𝐴𝐵 → (𝐵𝐶𝐴𝐶))
32con3dimp 407 1 ((𝐴𝐵 ∧ ¬ 𝐴𝐶) → ¬ 𝐵𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 394  wcel 2098  wss 3944
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100
This theorem depends on definitions:  df-bi 206  df-an 395  df-ex 1774  df-clel 2802  df-ss 3961
This theorem is referenced by:  nrelv  5802  ordtr3  6416  smndex2dnrinv  18875  frlmssuvc2  21746  1arithidom  33349  tfsconcatb0  42915  clsk1indlem1  43617  mapssbi  44725  fourierdlem10  45643  salgensscntex  45870
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