ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  soirri GIF version

Theorem soirri 4903
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 id 19 . 2 (𝐴𝑅𝐴𝐴𝑅𝐴)
2 soi.1 . . 3 𝑅 Or 𝑆
3 soi.2 . . . . 5 𝑅 ⊆ (𝑆 × 𝑆)
43brel 4561 . . . 4 (𝐴𝑅𝐴 → (𝐴𝑆𝐴𝑆))
54simpld 111 . . 3 (𝐴𝑅𝐴𝐴𝑆)
6 sonr 4209 . . 3 ((𝑅 Or 𝑆𝐴𝑆) → ¬ 𝐴𝑅𝐴)
72, 5, 6sylancr 410 . 2 (𝐴𝑅𝐴 → ¬ 𝐴𝑅𝐴)
81, 7pm2.65i 613 1 ¬ 𝐴𝑅𝐴
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wcel 1465  wss 3041   class class class wbr 3899   Or wor 4187   × cxp 4507
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-opab 3960  df-po 4188  df-iso 4189  df-xp 4515
This theorem is referenced by:  son2lpi  4905  ltsonq  7174  genpdisj  7299  ltposr  7539  axpre-ltirr  7658
  Copyright terms: Public domain W3C validator