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Theorem ssnpss 4104
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 4087 . . 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 3949  wpss 3950
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 2942  df-v 3477  df-in 3956  df-ss 3966  df-pss 3968
This theorem is referenced by:  npss0  4446  sorpssuni  7722  sorpssint  7723  suplem2pr  11048  symgvalstruct  19264  symgvalstructOLD  19265  lsppratlem6  20765  atcvati  31639  finxpreclem3  36274  lsatcvat  37920
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