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Theorem nelss 4029
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 3960 . . 3 (𝐵𝐶 → (𝐴𝐵𝐴𝐶))
21com12 32 . 2 (𝐴𝐵 → (𝐵𝐶𝐴𝐶))
32con3dimp 409 1 ((𝐴𝐵 ∧ ¬ 𝐴𝐶) → ¬ 𝐵𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  wcel 2105  wss 3935
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2793
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2800  df-cleq 2814  df-clel 2893  df-in 3942  df-ss 3951
This theorem is referenced by:  nrelv  5667  ordtr3  6230  frlmssuvc2  20869  clsk1indlem1  40275  mapssbi  41356  fourierdlem10  42283  salgensscntex  42508  smndex2dnrinv  43985
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