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Mirrors > Home > ILE Home > Th. List > pitri3or | Unicode version |
Description: Trichotomy for positive integers. (Contributed by Jim Kingdon, 21-Sep-2019.) |
Ref | Expression |
---|---|
pitri3or |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pinn 7085 | . . 3 | |
2 | pinn 7085 | . . 3 | |
3 | nntri3or 6357 | . . 3 | |
4 | 1, 2, 3 | syl2an 287 | . 2 |
5 | ltpiord 7095 | . . 3 | |
6 | biidd 171 | . . 3 | |
7 | ltpiord 7095 | . . . 4 | |
8 | 7 | ancoms 266 | . . 3 |
9 | 5, 6, 8 | 3orbi123d 1274 | . 2 |
10 | 4, 9 | mpbird 166 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3o 946 wceq 1316 wcel 1465 class class class wbr 3899 com 4474 cnpi 7048 clti 7051 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-nul 4024 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-iinf 4472 |
This theorem depends on definitions: df-bi 116 df-3or 948 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-br 3900 df-opab 3960 df-tr 3997 df-eprel 4181 df-iord 4258 df-on 4260 df-suc 4263 df-iom 4475 df-xp 4515 df-ni 7080 df-lti 7083 |
This theorem is referenced by: nqtri3or 7172 caucvgprlemnkj 7442 caucvgprlemnbj 7443 caucvgprprlemnkj 7468 caucvgprprlemnbj 7469 caucvgsr 7578 |
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