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Theorem soirri 4813
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 4478 . . . 4  |-  ( A R A  ->  ( A  e.  S  /\  A  e.  S )
)
54simpld 110 . . 3  |-  ( A R A  ->  A  e.  S )
6 sonr 4135 . . 3  |-  ( ( R  Or  S  /\  A  e.  S )  ->  -.  A R A )
72, 5, 6sylancr 405 . 2  |-  ( A R A  ->  -.  A R A )
81, 7pm2.65i 603 1  |-  -.  A R A
Colors of variables: wff set class
Syntax hints:   -. wn 3    e. wcel 1438    C_ wss 2997   class class class wbr 3837    Or wor 4113    X. cxp 4426
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-br 3838  df-opab 3892  df-po 4114  df-iso 4115  df-xp 4434
This theorem is referenced by:  son2lpi  4815  ltsonq  6936  genpdisj  7061  ltposr  7288  axpre-ltirr  7396
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