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Theorem soirri 5041
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 4696 . . . 4 (𝐴𝑅𝐴 → (𝐴𝑆𝐴𝑆))
54simpld 112 . . 3 (𝐴𝑅𝐴𝐴𝑆)
6 sonr 4335 . . 3 ((𝑅 Or 𝑆𝐴𝑆) → ¬ 𝐴𝑅𝐴)
72, 5, 6sylancr 414 . 2 (𝐴𝑅𝐴 → ¬ 𝐴𝑅𝐴)
81, 7pm2.65i 640 1 ¬ 𝐴𝑅𝐴
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wcel 2160  wss 3144   class class class wbr 4018   Or wor 4313   × cxp 4642
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019  df-opab 4080  df-po 4314  df-iso 4315  df-xp 4650
This theorem is referenced by:  son2lpi  5043  ltsonq  7427  genpdisj  7552  ltposr  7792  axpre-ltirr  7911
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