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Theorem soirri 4780
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 4447 . . . 4 (𝐴𝑅𝐴 → (𝐴𝑆𝐴𝑆))
54simpld 110 . . 3 (𝐴𝑅𝐴𝐴𝑆)
6 sonr 4107 . . 3 ((𝑅 Or 𝑆𝐴𝑆) → ¬ 𝐴𝑅𝐴)
72, 5, 6sylancr 405 . 2 (𝐴𝑅𝐴 → ¬ 𝐴𝑅𝐴)
81, 7pm2.65i 601 1 ¬ 𝐴𝑅𝐴
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wcel 1434  wss 2984   class class class wbr 3811   Or wor 4085   × cxp 4398
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-pow 3974  ax-pr 3999
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2614  df-un 2988  df-in 2990  df-ss 2997  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-br 3812  df-opab 3866  df-po 4086  df-iso 4087  df-xp 4406
This theorem is referenced by:  son2lpi  4782  ltsonq  6859  genpdisj  6984  ltposr  7211  axpre-ltirr  7319
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