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Theorem issod 4056
 Description: An irreflexive, transitive, trichotomous relation is a linear ordering (in the sense of df-iso 4034). (Contributed by NM, 21-Jan-1996.) (Revised by Mario Carneiro, 9-Jul-2014.)
Hypotheses
Ref Expression
issod.1
issod.2
Assertion
Ref Expression
issod
Distinct variable groups:   ,,   ,,   ,,

Proof of Theorem issod
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 issod.1 . 2
2 issod.2 . . . . . . . . . . 11
323adant3 924 . . . . . . . . . 10
4 orc 633 . . . . . . . . . . . 12
54a1i 9 . . . . . . . . . . 11
6 simp3r 933 . . . . . . . . . . . . 13
7 breq1 3767 . . . . . . . . . . . . 13
86, 7syl5ibcom 144 . . . . . . . . . . . 12
9 olc 632 . . . . . . . . . . . 12
108, 9syl6 29 . . . . . . . . . . 11
11 simp1 904 . . . . . . . . . . . . 13
12 simp2r 931 . . . . . . . . . . . . . 14
13 simp2l 930 . . . . . . . . . . . . . 14
14 simp3l 932 . . . . . . . . . . . . . 14
1512, 13, 143jca 1084 . . . . . . . . . . . . 13
16 potr 4045 . . . . . . . . . . . . . . . 16
171, 16sylan 267 . . . . . . . . . . . . . . 15
1817expcomd 1330 . . . . . . . . . . . . . 14
1918imp 115 . . . . . . . . . . . . 13
2011, 15, 6, 19syl21anc 1134 . . . . . . . . . . . 12
2120, 9syl6 29 . . . . . . . . . . 11
225, 10, 213jaod 1199 . . . . . . . . . 10
233, 22mpd 13 . . . . . . . . 9
24233expa 1104 . . . . . . . 8
2524expr 357 . . . . . . 7
2625ralrimiva 2392 . . . . . 6
2726anassrs 380 . . . . 5
2827ralrimiva 2392 . . . 4
29 ralcom 2473 . . . 4
3028, 29sylib 127 . . 3
3130ralrimiva 2392 . 2
32 df-iso 4034 . 2
331, 31, 32sylanbrc 394 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 97   wo 629   w3o 884   w3a 885   wcel 1393  wral 2306   class class class wbr 3764   wpo 4031   wor 4032 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-3or 886  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-v 2559  df-un 2922  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-po 4033  df-iso 4034 This theorem is referenced by:  ltsopi  6418  ltsonq  6496
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