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Mirrors > Home > ILE Home > Th. List > nntri1 | Unicode version |
Description: A trichotomy law for natural numbers. (Contributed by Jim Kingdon, 28-Aug-2019.) |
Ref | Expression |
---|---|
nntri1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssnel 4454 | . 2 | |
2 | nntri3or 6357 | . . . 4 | |
3 | df-3or 948 | . . . . . . 7 | |
4 | 3 | biimpi 119 | . . . . . 6 |
5 | 4 | orcomd 703 | . . . . 5 |
6 | 5 | ord 698 | . . . 4 |
7 | 2, 6 | syl 14 | . . 3 |
8 | nnord 4495 | . . . . . . 7 | |
9 | ordelss 4271 | . . . . . . 7 | |
10 | 8, 9 | sylan 281 | . . . . . 6 |
11 | 10 | ex 114 | . . . . 5 |
12 | 11 | adantl 275 | . . . 4 |
13 | eqimss 3121 | . . . . 5 | |
14 | 13 | a1i 9 | . . . 4 |
15 | 12, 14 | jaod 691 | . . 3 |
16 | 7, 15 | syld 45 | . 2 |
17 | 1, 16 | impbid2 142 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wo 682 w3o 946 wceq 1316 wcel 1465 wss 3041 word 4254 com 4474 |
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-setind 4422 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-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 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-uni 3707 df-int 3742 df-tr 3997 df-iord 4258 df-on 4260 df-suc 4263 df-iom 4475 |
This theorem is referenced by: nnsseleq 6365 nnmword 6382 nnawordex 6392 nndomo 6726 ennnfonelemex 11854 pwle2 13120 nninfalllemn 13129 |
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