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Mirrors > Home > ILE Home > Th. List > nntri3or | Unicode version |
Description: Trichotomy for natural numbers. (Contributed by Jim Kingdon, 25-Aug-2019.) |
Ref | Expression |
---|---|
nntri3or |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq2 2203 | . . . . 5 | |
2 | eqeq2 2149 | . . . . 5 | |
3 | eleq1 2202 | . . . . 5 | |
4 | 1, 2, 3 | 3orbi123d 1289 | . . . 4 |
5 | 4 | imbi2d 229 | . . 3 |
6 | eleq2 2203 | . . . . 5 | |
7 | eqeq2 2149 | . . . . 5 | |
8 | eleq1 2202 | . . . . 5 | |
9 | 6, 7, 8 | 3orbi123d 1289 | . . . 4 |
10 | eleq2 2203 | . . . . 5 | |
11 | eqeq2 2149 | . . . . 5 | |
12 | eleq1 2202 | . . . . 5 | |
13 | 10, 11, 12 | 3orbi123d 1289 | . . . 4 |
14 | eleq2 2203 | . . . . 5 | |
15 | eqeq2 2149 | . . . . 5 | |
16 | eleq1 2202 | . . . . 5 | |
17 | 14, 15, 16 | 3orbi123d 1289 | . . . 4 |
18 | 0elnn 4532 | . . . . 5 | |
19 | olc 700 | . . . . . 6 | |
20 | 3orass 965 | . . . . . 6 | |
21 | 19, 20 | sylibr 133 | . . . . 5 |
22 | 18, 21 | syl 14 | . . . 4 |
23 | df-3or 963 | . . . . . 6 | |
24 | elex 2697 | . . . . . . . 8 | |
25 | elsuc2g 4327 | . . . . . . . . 9 | |
26 | 3mix1 1150 | . . . . . . . . 9 | |
27 | 25, 26 | syl6bir 163 | . . . . . . . 8 |
28 | 24, 27 | syl 14 | . . . . . . 7 |
29 | nnsucelsuc 6387 | . . . . . . . . 9 | |
30 | elsuci 4325 | . . . . . . . . 9 | |
31 | 29, 30 | syl6bi 162 | . . . . . . . 8 |
32 | eqcom 2141 | . . . . . . . . . . . . 13 | |
33 | 32 | orbi2i 751 | . . . . . . . . . . . 12 |
34 | 33 | biimpi 119 | . . . . . . . . . . 11 |
35 | 34 | orcomd 718 | . . . . . . . . . 10 |
36 | 35 | olcd 723 | . . . . . . . . 9 |
37 | 3orass 965 | . . . . . . . . 9 | |
38 | 36, 37 | sylibr 133 | . . . . . . . 8 |
39 | 31, 38 | syl6 33 | . . . . . . 7 |
40 | 28, 39 | jaao 708 | . . . . . 6 |
41 | 23, 40 | syl5bi 151 | . . . . 5 |
42 | 41 | ex 114 | . . . 4 |
43 | 9, 13, 17, 22, 42 | finds2 4515 | . . 3 |
44 | 5, 43 | vtoclga 2752 | . 2 |
45 | 44 | impcom 124 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wo 697 w3o 961 wceq 1331 wcel 1480 cvv 2686 c0 3363 csuc 4287 com 4504 |
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-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-iinf 4502 |
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-rex 2422 df-v 2688 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-uni 3737 df-int 3772 df-tr 4027 df-iord 4288 df-on 4290 df-suc 4293 df-iom 4505 |
This theorem is referenced by: nntri2 6390 nntri1 6392 nntri3 6393 nntri2or2 6394 nndceq 6395 nndcel 6396 nnsseleq 6397 nntr2 6399 nnawordex 6424 nnwetri 6804 ltsopi 7128 pitri3or 7130 frec2uzlt2d 10177 ennnfonelemk 11913 ennnfonelemex 11927 nninfalllemn 13202 |
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