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Theorem nelss 3980
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 3909 . . 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 2112  wss 3882
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2114  ax-9 2122  ax-ext 2710
This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1546  df-ex 1788  df-sb 2073  df-clab 2717  df-cleq 2731  df-clel 2818  df-v 3424  df-in 3889  df-ss 3899
This theorem is referenced by:  nrelv  5687  ordtr3  6278  smndex2dnrinv  18372  frlmssuvc2  20787  clsk1indlem1  41367  mapssbi  42461  fourierdlem10  43366  salgensscntex  43591
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