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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 2181 | . . . . 5 | |
2 | eqeq2 2127 | . . . . 5 | |
3 | eleq1 2180 | . . . . 5 | |
4 | 1, 2, 3 | 3orbi123d 1274 | . . . 4 |
5 | 4 | imbi2d 229 | . . 3 |
6 | eleq2 2181 | . . . . 5 | |
7 | eqeq2 2127 | . . . . 5 | |
8 | eleq1 2180 | . . . . 5 | |
9 | 6, 7, 8 | 3orbi123d 1274 | . . . 4 |
10 | eleq2 2181 | . . . . 5 | |
11 | eqeq2 2127 | . . . . 5 | |
12 | eleq1 2180 | . . . . 5 | |
13 | 10, 11, 12 | 3orbi123d 1274 | . . . 4 |
14 | eleq2 2181 | . . . . 5 | |
15 | eqeq2 2127 | . . . . 5 | |
16 | eleq1 2180 | . . . . 5 | |
17 | 14, 15, 16 | 3orbi123d 1274 | . . . 4 |
18 | 0elnn 4502 | . . . . 5 | |
19 | olc 685 | . . . . . 6 | |
20 | 3orass 950 | . . . . . 6 | |
21 | 19, 20 | sylibr 133 | . . . . 5 |
22 | 18, 21 | syl 14 | . . . 4 |
23 | df-3or 948 | . . . . . 6 | |
24 | elex 2671 | . . . . . . . 8 | |
25 | elsuc2g 4297 | . . . . . . . . 9 | |
26 | 3mix1 1135 | . . . . . . . . 9 | |
27 | 25, 26 | syl6bir 163 | . . . . . . . 8 |
28 | 24, 27 | syl 14 | . . . . . . 7 |
29 | nnsucelsuc 6355 | . . . . . . . . 9 | |
30 | elsuci 4295 | . . . . . . . . 9 | |
31 | 29, 30 | syl6bi 162 | . . . . . . . 8 |
32 | eqcom 2119 | . . . . . . . . . . . . 13 | |
33 | 32 | orbi2i 736 | . . . . . . . . . . . 12 |
34 | 33 | biimpi 119 | . . . . . . . . . . 11 |
35 | 34 | orcomd 703 | . . . . . . . . . 10 |
36 | 35 | olcd 708 | . . . . . . . . 9 |
37 | 3orass 950 | . . . . . . . . 9 | |
38 | 36, 37 | sylibr 133 | . . . . . . . 8 |
39 | 31, 38 | syl6 33 | . . . . . . 7 |
40 | 28, 39 | jaao 693 | . . . . . 6 |
41 | 23, 40 | syl5bi 151 | . . . . 5 |
42 | 41 | ex 114 | . . . 4 |
43 | 9, 13, 17, 22, 42 | finds2 4485 | . . 3 |
44 | 5, 43 | vtoclga 2726 | . 2 |
45 | 44 | impcom 124 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wo 682 w3o 946 wceq 1316 wcel 1465 cvv 2660 c0 3333 csuc 4257 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-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-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: nntri2 6358 nntri1 6360 nntri3 6361 nntri2or2 6362 nndceq 6363 nndcel 6364 nnsseleq 6365 nntr2 6367 nnawordex 6392 nnwetri 6772 ltsopi 7096 pitri3or 7098 frec2uzlt2d 10132 ennnfonelemk 11824 ennnfonelemex 11838 nninfalllemn 13098 |
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