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Theorem ssnpss 4064
Description: Partial trichotomy law for subclasses. (Contributed by NM, 16-May-1996.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
ssnpss (𝐴𝐵 → ¬ 𝐵𝐴)

Proof of Theorem ssnpss
StepHypRef Expression
1 dfpss3 4047 . . 3 (𝐵𝐴 ↔ (𝐵𝐴 ∧ ¬ 𝐴𝐵))
21simprbi 498 . 2 (𝐵𝐴 → ¬ 𝐴𝐵)
32con2i 139 1 (𝐴𝐵 → ¬ 𝐵𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wss 3911  wpss 3912
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2941  df-v 3446  df-in 3918  df-ss 3928  df-pss 3930
This theorem is referenced by:  npss0  4406  sorpssuni  7670  sorpssint  7671  suplem2pr  10994  symgvalstruct  19183  symgvalstructOLD  19184  lsppratlem6  20629  atcvati  31370  finxpreclem3  35910  lsatcvat  37558
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