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Theorem soirri 4869
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  C_  ( S  X.  S
)
Assertion
Ref Expression
soirri  |-  -.  A R A

Proof of Theorem soirri
StepHypRef Expression
1 id 19 . 2  |-  ( A R A  ->  A R A )
2 soi.1 . . 3  |-  R  Or  S
3 soi.2 . . . . 5  |-  R  C_  ( S  X.  S
)
43brel 4529 . . . 4  |-  ( A R A  ->  ( A  e.  S  /\  A  e.  S )
)
54simpld 111 . . 3  |-  ( A R A  ->  A  e.  S )
6 sonr 4177 . . 3  |-  ( ( R  Or  S  /\  A  e.  S )  ->  -.  A R A )
72, 5, 6sylancr 408 . 2  |-  ( A R A  ->  -.  A R A )
81, 7pm2.65i 608 1  |-  -.  A R A
Colors of variables: wff set class
Syntax hints:   -. wn 3    e. wcel 1448    C_ wss 3021   class class class wbr 3875    Or wor 4155    X. cxp 4475
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069
This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-v 2643  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-br 3876  df-opab 3930  df-po 4156  df-iso 4157  df-xp 4483
This theorem is referenced by:  son2lpi  4871  ltsonq  7107  genpdisj  7232  ltposr  7459  axpre-ltirr  7567
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