ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  issod Unicode version

Theorem issod 4047
Description: An irreflexive, transitive, trichotomous relation is a linear ordering (in the sense of df-iso 4025). (Contributed by NM, 21-Jan-1996.) (Revised by Mario Carneiro, 9-Jul-2014.)
Hypotheses
Ref Expression
issod.1  R  Po
issod.2  R  R
Assertion
Ref Expression
issod  R  Or
Distinct variable groups:   ,, R   ,,   ,,

Proof of Theorem issod
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 issod.1 . 2  R  Po
2 issod.2 . . . . . . . . . . 11  R  R
323adant3 923 . . . . . . . . . 10  R  R  R
4 orc 632 . . . . . . . . . . . 12  R  R  R
54a1i 9 . . . . . . . . . . 11  R  R  R  R
6 simp3r 932 . . . . . . . . . . . . 13  R  R
7 breq1 3758 . . . . . . . . . . . . 13  R  R
86, 7syl5ibcom 144 . . . . . . . . . . . 12  R  R
9 olc 631 . . . . . . . . . . . 12  R  R  R
108, 9syl6 29 . . . . . . . . . . 11  R  R  R
11 simp1 903 . . . . . . . . . . . . 13  R
12 simp2r 930 . . . . . . . . . . . . . 14  R
13 simp2l 929 . . . . . . . . . . . . . 14  R
14 simp3l 931 . . . . . . . . . . . . . 14  R
1512, 13, 143jca 1083 . . . . . . . . . . . . 13  R
16 potr 4036 . . . . . . . . . . . . . . . 16  R  Po  R  R  R
171, 16sylan 267 . . . . . . . . . . . . . . 15  R  R  R
1817expcomd 1327 . . . . . . . . . . . . . 14  R  R  R
1918imp 115 . . . . . . . . . . . . 13  R  R  R
2011, 15, 6, 19syl21anc 1133 . . . . . . . . . . . 12  R  R  R
2120, 9syl6 29 . . . . . . . . . . 11  R  R  R  R
225, 10, 213jaod 1198 . . . . . . . . . 10  R  R  R  R  R
233, 22mpd 13 . . . . . . . . 9  R  R  R
24233expa 1103 . . . . . . . 8  R  R  R
2524expr 357 . . . . . . 7  R  R  R
2625ralrimiva 2386 . . . . . 6  R  R  R
2726anassrs 380 . . . . 5  R  R  R
2827ralrimiva 2386 . . . 4  R  R  R
29 ralcom 2467 . . . 4  R  R  R  R  R  R
3028, 29sylib 127 . . 3  R  R  R
3130ralrimiva 2386 . 2  R  R  R
32 df-iso 4025 . 2  R  Or  R  Po  R  R  R
331, 31, 32sylanbrc 394 1  R  Or
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wo 628   w3o 883   w3a 884   wcel 1390  wral 2300   class class class wbr 3755    Po wpo 4022    Or wor 4023
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-3or 885  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-v 2553  df-un 2916  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-po 4024  df-iso 4025
This theorem is referenced by:  ltsopi  6304  ltsonq  6382
  Copyright terms: Public domain W3C validator