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Theorem sotrieq 4832
Description: Trichotomy law for strict order relation. (Contributed by NM, 9-Apr-1996.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
Assertion
Ref Expression
sotrieq

Proof of Theorem sotrieq
StepHypRef Expression
1 sonr 4826 . . . . . . 7
21adantrr 716 . . . . . 6
3 pm1.2 513 . . . . . 6
42, 3nsyl 121 . . . . 5
5 breq2 4456 . . . . . . 7
6 breq1 4455 . . . . . . 7
75, 6orbi12d 709 . . . . . 6
87notbid 294 . . . . 5
94, 8syl5ibcom 220 . . . 4
109con2d 115 . . 3
11 solin 4828 . . . 4
12 3orass 976 . . . . 5
13 or12 523 . . . . 5
14 df-or 370 . . . . 5
1512, 13, 143bitri 271 . . . 4
1611, 15sylib 196 . . 3
1710, 16impbid 191 . 2
1817con2bid 329 1
Colors of variables: wff setvar class
Syntax hints:  -.wn 3  ->wi 4  <->wb 184  \/wo 368  /\wa 369  \/w3o 972  =wceq 1395  e.wcel 1818   class class class wbr 4452  Orwor 4804
This theorem is referenced by:  sotrieq2  4833  sossfld  5459  soisores  6223  soisoi  6224  weniso  6250  wemapsolem  7996  distrlem4pr  9425  addcanpr  9445  sqgt0sr  9504  lttri2  9688  xrlttri2  11377  xrltne  11395  soseq  29334
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812  df-rab 2816  df-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-br 4453  df-po 4805  df-so 4806
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