Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  ssnpss Structured version   Visualization version   GIF version

Theorem ssnpss 3672
 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 3655 . . 3 (𝐵𝐴 ↔ (𝐵𝐴 ∧ ¬ 𝐴𝐵))
21simprbi 479 . 2 (𝐵𝐴 → ¬ 𝐴𝐵)
32con2i 133 1 (𝐴𝐵 → ¬ 𝐵𝐴)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ⊆ wss 3540   ⊊ wpss 3541 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-ne 2782  df-in 3547  df-ss 3554  df-pss 3556 This theorem is referenced by:  npss0  3966  sorpssuni  6844  sorpssint  6845  suplem2pr  9754  lsppratlem6  18973  atcvati  28629  finxpreclem3  32406  lsatcvat  33355
 Copyright terms: Public domain W3C validator