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Theorem ssnpss 3992
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 3975 . . 3 (𝐵𝐴 ↔ (𝐵𝐴 ∧ ¬ 𝐴𝐵))
21simprbi 500 . 2 (𝐵𝐴 → ¬ 𝐴𝐵)
32con2i 141 1 (𝐴𝐵 → ¬ 𝐵𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wss 3841  wpss 3842
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-ext 2710
This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1545  df-ex 1787  df-sb 2074  df-clab 2717  df-cleq 2730  df-clel 2811  df-ne 2935  df-v 3399  df-in 3848  df-ss 3858  df-pss 3860
This theorem is referenced by:  npss0  4332  sorpssuni  7470  sorpssint  7471  suplem2pr  10546  symgvalstruct  18636  lsppratlem6  20036  atcvati  30313  finxpreclem3  35176  lsatcvat  36676
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