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Theorem soirri 4933
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 4591 . . . 4 (𝐴𝑅𝐴 → (𝐴𝑆𝐴𝑆))
54simpld 111 . . 3 (𝐴𝑅𝐴𝐴𝑆)
6 sonr 4239 . . 3 ((𝑅 Or 𝑆𝐴𝑆) → ¬ 𝐴𝑅𝐴)
72, 5, 6sylancr 410 . 2 (𝐴𝑅𝐴 → ¬ 𝐴𝑅𝐴)
81, 7pm2.65i 628 1 ¬ 𝐴𝑅𝐴
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wcel 1480  wss 3071   class class class wbr 3929   Or wor 4217   × cxp 4537
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-po 4218  df-iso 4219  df-xp 4545
This theorem is referenced by:  son2lpi  4935  ltsonq  7218  genpdisj  7343  ltposr  7583  axpre-ltirr  7702
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