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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 4219). (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 1001 | . . . . . . . . . 10 |
4 | orc 701 | . . . . . . . . . . . 12 | |
5 | 4 | a1i 9 | . . . . . . . . . . 11 |
6 | simp3r 1010 | . . . . . . . . . . . . 13 | |
7 | breq1 3932 | . . . . . . . . . . . . 13 | |
8 | 6, 7 | syl5ibcom 154 | . . . . . . . . . . . 12 |
9 | olc 700 | . . . . . . . . . . . 12 | |
10 | 8, 9 | syl6 33 | . . . . . . . . . . 11 |
11 | simp1 981 | . . . . . . . . . . . . 13 | |
12 | simp2r 1008 | . . . . . . . . . . . . . 14 | |
13 | simp2l 1007 | . . . . . . . . . . . . . 14 | |
14 | simp3l 1009 | . . . . . . . . . . . . . 14 | |
15 | 12, 13, 14 | 3jca 1161 | . . . . . . . . . . . . 13 |
16 | potr 4230 | . . . . . . . . . . . . . . . 16 | |
17 | 1, 16 | sylan 281 | . . . . . . . . . . . . . . 15 |
18 | 17 | expcomd 1417 | . . . . . . . . . . . . . 14 |
19 | 18 | imp 123 | . . . . . . . . . . . . 13 |
20 | 11, 15, 6, 19 | syl21anc 1215 | . . . . . . . . . . . 12 |
21 | 20, 9 | syl6 33 | . . . . . . . . . . 11 |
22 | 5, 10, 21 | 3jaod 1282 | . . . . . . . . . 10 |
23 | 3, 22 | mpd 13 | . . . . . . . . 9 |
24 | 23 | 3expa 1181 | . . . . . . . 8 |
25 | 24 | expr 372 | . . . . . . 7 |
26 | 25 | ralrimiva 2505 | . . . . . 6 |
27 | 26 | anassrs 397 | . . . . 5 |
28 | 27 | ralrimiva 2505 | . . . 4 |
29 | ralcom 2594 | . . . 4 | |
30 | 28, 29 | sylib 121 | . . 3 |
31 | 30 | ralrimiva 2505 | . 2 |
32 | df-iso 4219 | . 2 | |
33 | 1, 31, 32 | sylanbrc 413 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wo 697 w3o 961 w3a 962 wcel 1480 wral 2416 class class class wbr 3929 wpo 4216 wor 4217 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-3or 963 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-v 2688 df-un 3075 df-sn 3533 df-pr 3534 df-op 3536 df-br 3930 df-po 4218 df-iso 4219 |
This theorem is referenced by: ltsopi 7128 ltsonq 7206 |
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