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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 4052). (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 958 | . . . . . . . . . 10 |
4 | orc 665 | . . . . . . . . . . . 12 | |
5 | 4 | a1i 9 | . . . . . . . . . . 11 |
6 | simp3r 967 | . . . . . . . . . . . . 13 | |
7 | breq1 3788 | . . . . . . . . . . . . 13 | |
8 | 6, 7 | syl5ibcom 153 | . . . . . . . . . . . 12 |
9 | olc 664 | . . . . . . . . . . . 12 | |
10 | 8, 9 | syl6 33 | . . . . . . . . . . 11 |
11 | simp1 938 | . . . . . . . . . . . . 13 | |
12 | simp2r 965 | . . . . . . . . . . . . . 14 | |
13 | simp2l 964 | . . . . . . . . . . . . . 14 | |
14 | simp3l 966 | . . . . . . . . . . . . . 14 | |
15 | 12, 13, 14 | 3jca 1118 | . . . . . . . . . . . . 13 |
16 | potr 4063 | . . . . . . . . . . . . . . . 16 | |
17 | 1, 16 | sylan 277 | . . . . . . . . . . . . . . 15 |
18 | 17 | expcomd 1370 | . . . . . . . . . . . . . 14 |
19 | 18 | imp 122 | . . . . . . . . . . . . 13 |
20 | 11, 15, 6, 19 | syl21anc 1168 | . . . . . . . . . . . 12 |
21 | 20, 9 | syl6 33 | . . . . . . . . . . 11 |
22 | 5, 10, 21 | 3jaod 1235 | . . . . . . . . . 10 |
23 | 3, 22 | mpd 13 | . . . . . . . . 9 |
24 | 23 | 3expa 1138 | . . . . . . . 8 |
25 | 24 | expr 367 | . . . . . . 7 |
26 | 25 | ralrimiva 2434 | . . . . . 6 |
27 | 26 | anassrs 392 | . . . . 5 |
28 | 27 | ralrimiva 2434 | . . . 4 |
29 | ralcom 2517 | . . . 4 | |
30 | 28, 29 | sylib 120 | . . 3 |
31 | 30 | ralrimiva 2434 | . 2 |
32 | df-iso 4052 | . 2 | |
33 | 1, 31, 32 | sylanbrc 408 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wo 661 w3o 918 w3a 919 wcel 1433 wral 2348 class class class wbr 3785 wpo 4049 wor 4050 |
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 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-3or 920 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-v 2603 df-un 2977 df-sn 3404 df-pr 3405 df-op 3407 df-br 3786 df-po 4051 df-iso 4052 |
This theorem is referenced by: ltsopi 6510 ltsonq 6588 |
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