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Theorem soirri 6084
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 5572 . . . 4 ((𝑅 Or 𝑆𝐴𝑆) → ¬ 𝐴𝑅𝐴)
31, 2mpan 689 . . 3 (𝐴𝑆 → ¬ 𝐴𝑅𝐴)
43adantl 483 . 2 ((𝐴𝑆𝐴𝑆) → ¬ 𝐴𝑅𝐴)
5 soi.2 . . . 4 𝑅 ⊆ (𝑆 × 𝑆)
65brel 5701 . . 3 (𝐴𝑅𝐴 → (𝐴𝑆𝐴𝑆))
76con3i 154 . 2 (¬ (𝐴𝑆𝐴𝑆) → ¬ 𝐴𝑅𝐴)
84, 7pm2.61i 182 1 ¬ 𝐴𝑅𝐴
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 397  wcel 2107  wss 3914   class class class wbr 5109   Or wor 5548   × cxp 5635
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-br 5110  df-opab 5172  df-po 5549  df-so 5550  df-xp 5643
This theorem is referenced by:  son2lpi  6086  nqpr  10958  ltapr  10989
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