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Theorem ssnpss 4078
 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 4061 . . 3 (𝐵𝐴 ↔ (𝐵𝐴 ∧ ¬ 𝐴𝐵))
21simprbi 499 . 2 (𝐵𝐴 → ¬ 𝐴𝐵)
32con2i 141 1 (𝐴𝐵 → ¬ 𝐵𝐴)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ⊆ wss 3934   ⊊ wpss 3935 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-clab 2798  df-cleq 2812  df-clel 2891  df-ne 3015  df-in 3941  df-ss 3950  df-pss 3952 This theorem is referenced by:  npss0  4395  sorpssuni  7450  sorpssint  7451  suplem2pr  10467  symgvalstruct  18517  lsppratlem6  19916  atcvati  30155  finxpreclem3  34666  lsatcvat  36178
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