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Theorem soirri 5979
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 5489 . . . 4 ((𝑅 Or 𝑆𝐴𝑆) → ¬ 𝐴𝑅𝐴)
31, 2mpan 686 . . 3 (𝐴𝑆 → ¬ 𝐴𝑅𝐴)
43adantl 482 . 2 ((𝐴𝑆𝐴𝑆) → ¬ 𝐴𝑅𝐴)
5 soi.2 . . . 4 𝑅 ⊆ (𝑆 × 𝑆)
65brel 5610 . . 3 (𝐴𝑅𝐴 → (𝐴𝑆𝐴𝑆))
76con3i 157 . 2 (¬ (𝐴𝑆𝐴𝑆) → ¬ 𝐴𝑅𝐴)
84, 7pm2.61i 183 1 ¬ 𝐴𝑅𝐴
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 396  wcel 2105  wss 3933   class class class wbr 5057   Or wor 5466   × cxp 5546
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120  df-po 5467  df-so 5468  df-xp 5554
This theorem is referenced by:  son2lpi  5981  nqpr  10424  ltapr  10455
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