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

Theorem pssirr 3664
Description: Proper subclass is irreflexive. Theorem 7 of [Suppes] p. 23. (Contributed by NM, 7-Feb-1996.)
Assertion
Ref Expression
pssirr ¬ 𝐴𝐴

Proof of Theorem pssirr
StepHypRef Expression
1 pm3.24 921 . 2 ¬ (𝐴𝐴 ∧ ¬ 𝐴𝐴)
2 dfpss3 3650 . 2 (𝐴𝐴 ↔ (𝐴𝐴 ∧ ¬ 𝐴𝐴))
31, 2mtbir 311 1 ¬ 𝐴𝐴
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 382  wss 3535  wpss 3536
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-clab 2592  df-cleq 2598  df-clel 2601  df-ne 2777  df-in 3542  df-ss 3549  df-pss 3551
This theorem is referenced by:  porpss  6812  ltsopr  9706
  Copyright terms: Public domain W3C validator