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Mirrors > Home > ILE Home > Th. List > nntri3 | Unicode version |
Description: A trichotomy law for natural numbers. (Contributed by Jim Kingdon, 15-May-2020.) |
Ref | Expression |
---|---|
nntri3 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elirr 4456 | . . . . . 6 | |
2 | eleq2 2203 | . . . . . 6 | |
3 | 1, 2 | mtbii 663 | . . . . 5 |
4 | 3 | con2i 616 | . . . 4 |
5 | 4 | adantl 275 | . . 3 |
6 | simpl 108 | . . . . 5 | |
7 | 6 | con2i 616 | . . . 4 |
8 | 7 | adantl 275 | . . 3 |
9 | 5, 8 | 2falsed 691 | . 2 |
10 | simpr 109 | . . 3 | |
11 | eleq1 2202 | . . . . . 6 | |
12 | 1, 11 | mtbii 663 | . . . . 5 |
13 | 3, 12 | jca 304 | . . . 4 |
14 | 13 | adantl 275 | . . 3 |
15 | 10, 14 | 2thd 174 | . 2 |
16 | 12 | con2i 616 | . . . 4 |
17 | 16 | adantl 275 | . . 3 |
18 | simpr 109 | . . . . 5 | |
19 | 18 | con2i 616 | . . . 4 |
20 | 19 | adantl 275 | . . 3 |
21 | 17, 20 | 2falsed 691 | . 2 |
22 | nntri3or 6389 | . 2 | |
23 | 9, 15, 21, 22 | mpjao3dan 1285 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wceq 1331 wcel 1480 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-setind 4452 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-ne 2309 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: frec2uzf1od 10179 nnti 13191 |
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