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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 4530 | . 2 | |
2 | nntri3or 6442 | . . . 4 | |
3 | df-3or 964 | . . . . . . 7 | |
4 | 3 | biimpi 119 | . . . . . 6 |
5 | 4 | orcomd 719 | . . . . 5 |
6 | 5 | ord 714 | . . . 4 |
7 | 2, 6 | syl 14 | . . 3 |
8 | nnord 4573 | . . . . . . 7 | |
9 | ordelss 4341 | . . . . . . 7 | |
10 | 8, 9 | sylan 281 | . . . . . 6 |
11 | 10 | ex 114 | . . . . 5 |
12 | 11 | adantl 275 | . . . 4 |
13 | eqimss 3182 | . . . . 5 | |
14 | 13 | a1i 9 | . . . 4 |
15 | 12, 14 | jaod 707 | . . 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 698 w3o 962 wceq 1335 wcel 2128 wss 3102 word 4324 com 4551 |
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 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4084 ax-nul 4092 ax-pow 4137 ax-pr 4171 ax-un 4395 ax-setind 4498 ax-iinf 4549 |
This theorem depends on definitions: df-bi 116 df-3or 964 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-ral 2440 df-rex 2441 df-v 2714 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3396 df-pw 3546 df-sn 3567 df-pr 3568 df-uni 3775 df-int 3810 df-tr 4065 df-iord 4328 df-on 4330 df-suc 4333 df-iom 4552 |
This theorem is referenced by: nnsseleq 6450 nnmword 6467 nnawordex 6477 nndomo 6811 nnnninfeq 7073 ennnfonelemex 12213 pwle2 13641 |
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