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

Theorem soirri 5662
Description: A strict order relation is irreflexive. (Contributed by NM, 10-Feb-1996.) (Revised by Mario Carneiro, 10-May-2013.)
Hypotheses
Ref Expression
soi.1 𝑅 Or 𝑆
soi.2 𝑅 ⊆ (𝑆 × 𝑆)
Assertion
Ref Expression
soirri ¬ 𝐴𝑅𝐴

Proof of Theorem soirri
StepHypRef Expression
1 soi.1 . . . 4 𝑅 Or 𝑆
2 sonr 5192 . . . 4 ((𝑅 Or 𝑆𝐴𝑆) → ¬ 𝐴𝑅𝐴)
31, 2mpan 670 . . 3 (𝐴𝑆 → ¬ 𝐴𝑅𝐴)
43adantl 467 . 2 ((𝐴𝑆𝐴𝑆) → ¬ 𝐴𝑅𝐴)
5 soi.2 . . . 4 𝑅 ⊆ (𝑆 × 𝑆)
65brel 5307 . . 3 (𝐴𝑅𝐴 → (𝐴𝑆𝐴𝑆))
76con3i 151 . 2 (¬ (𝐴𝑆𝐴𝑆) → ¬ 𝐴𝑅𝐴)
84, 7pm2.61i 176 1 ¬ 𝐴𝑅𝐴
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 382  wcel 2145  wss 3723   class class class wbr 4787   Or wor 5170   × cxp 5248
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-br 4788  df-opab 4848  df-po 5171  df-so 5172  df-xp 5256
This theorem is referenced by:  son2lpi  5664  nqpr  10042  ltapr  10073
  Copyright terms: Public domain W3C validator