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Theorem soirri 5491
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 5026 . . . 4 ((𝑅 Or 𝑆𝐴𝑆) → ¬ 𝐴𝑅𝐴)
31, 2mpan 705 . . 3 (𝐴𝑆 → ¬ 𝐴𝑅𝐴)
43adantl 482 . 2 ((𝐴𝑆𝐴𝑆) → ¬ 𝐴𝑅𝐴)
5 soi.2 . . . 4 𝑅 ⊆ (𝑆 × 𝑆)
65brel 5138 . . 3 (𝐴𝑅𝐴 → (𝐴𝑆𝐴𝑆))
76con3i 150 . 2 (¬ (𝐴𝑆𝐴𝑆) → ¬ 𝐴𝑅𝐴)
84, 7pm2.61i 176 1 ¬ 𝐴𝑅𝐴
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 384  wcel 1987  wss 3560   class class class wbr 4623   Or wor 5004   × cxp 5082
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-br 4624  df-opab 4684  df-po 5005  df-so 5006  df-xp 5090
This theorem is referenced by:  son2lpi  5493  nqpr  9796  ltapr  9827
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