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Mirrors > Home > ILE Home > Th. List > issod | Unicode version |
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.) |
Ref | Expression |
---|---|
issod.1 | |
issod.2 |
Ref | Expression |
---|---|
issod |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | issod.1 | . 2 | |
2 | issod.2 | . . . . . . . . . . 11 | |
3 | 2 | 3adant3 924 | . . . . . . . . . 10 |
4 | orc 633 | . . . . . . . . . . . 12 | |
5 | 4 | a1i 9 | . . . . . . . . . . 11 |
6 | simp3r 933 | . . . . . . . . . . . . 13 | |
7 | breq1 3767 | . . . . . . . . . . . . 13 | |
8 | 6, 7 | syl5ibcom 144 | . . . . . . . . . . . 12 |
9 | olc 632 | . . . . . . . . . . . 12 | |
10 | 8, 9 | syl6 29 | . . . . . . . . . . 11 |
11 | simp1 904 | . . . . . . . . . . . . 13 | |
12 | simp2r 931 | . . . . . . . . . . . . . 14 | |
13 | simp2l 930 | . . . . . . . . . . . . . 14 | |
14 | simp3l 932 | . . . . . . . . . . . . . 14 | |
15 | 12, 13, 14 | 3jca 1084 | . . . . . . . . . . . . 13 |
16 | potr 4045 | . . . . . . . . . . . . . . . 16 | |
17 | 1, 16 | sylan 267 | . . . . . . . . . . . . . . 15 |
18 | 17 | expcomd 1330 | . . . . . . . . . . . . . 14 |
19 | 18 | imp 115 | . . . . . . . . . . . . 13 |
20 | 11, 15, 6, 19 | syl21anc 1134 | . . . . . . . . . . . 12 |
21 | 20, 9 | syl6 29 | . . . . . . . . . . 11 |
22 | 5, 10, 21 | 3jaod 1199 | . . . . . . . . . 10 |
23 | 3, 22 | mpd 13 | . . . . . . . . 9 |
24 | 23 | 3expa 1104 | . . . . . . . 8 |
25 | 24 | expr 357 | . . . . . . 7 |
26 | 25 | ralrimiva 2392 | . . . . . 6 |
27 | 26 | anassrs 380 | . . . . 5 |
28 | 27 | ralrimiva 2392 | . . . 4 |
29 | ralcom 2473 | . . . 4 | |
30 | 28, 29 | sylib 127 | . . 3 |
31 | 30 | ralrimiva 2392 | . 2 |
32 | df-iso 4034 | . 2 | |
33 | 1, 31, 32 | sylanbrc 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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