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Theorem soirri 4826
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 4490 . . . 4 (𝐴𝑅𝐴 → (𝐴𝑆𝐴𝑆))
54simpld 110 . . 3 (𝐴𝑅𝐴𝐴𝑆)
6 sonr 4144 . . 3 ((𝑅 Or 𝑆𝐴𝑆) → ¬ 𝐴𝑅𝐴)
72, 5, 6sylancr 405 . 2 (𝐴𝑅𝐴 → ¬ 𝐴𝑅𝐴)
81, 7pm2.65i 603 1 ¬ 𝐴𝑅𝐴
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wcel 1438  wss 2999   class class class wbr 3845   Or wor 4122   × cxp 4436
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-br 3846  df-opab 3900  df-po 4123  df-iso 4124  df-xp 4444
This theorem is referenced by:  son2lpi  4828  ltsonq  6957  genpdisj  7082  ltposr  7309  axpre-ltirr  7417
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