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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 4523 | . . . . . 6 | |
2 | eleq2 2234 | . . . . . 6 | |
3 | 1, 2 | mtbii 669 | . . . . 5 |
4 | 3 | con2i 622 | . . . 4 |
5 | 4 | adantl 275 | . . 3 |
6 | simpl 108 | . . . . 5 | |
7 | 6 | con2i 622 | . . . 4 |
8 | 7 | adantl 275 | . . 3 |
9 | 5, 8 | 2falsed 697 | . 2 |
10 | simpr 109 | . . 3 | |
11 | eleq1 2233 | . . . . . 6 | |
12 | 1, 11 | mtbii 669 | . . . . 5 |
13 | 3, 12 | jca 304 | . . . 4 |
14 | 13 | adantl 275 | . . 3 |
15 | 10, 14 | 2thd 174 | . 2 |
16 | 12 | con2i 622 | . . . 4 |
17 | 16 | adantl 275 | . . 3 |
18 | simpr 109 | . . . . 5 | |
19 | 18 | con2i 622 | . . . 4 |
20 | 19 | adantl 275 | . . 3 |
21 | 17, 20 | 2falsed 697 | . 2 |
22 | nntri3or 6469 | . 2 | |
23 | 9, 15, 21, 22 | mpjao3dan 1302 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wceq 1348 wcel 2141 com 4572 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-nul 4113 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 ax-iinf 4570 |
This theorem depends on definitions: df-bi 116 df-3or 974 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-v 2732 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3566 df-sn 3587 df-pr 3588 df-uni 3795 df-int 3830 df-tr 4086 df-iord 4349 df-on 4351 df-suc 4354 df-iom 4573 |
This theorem is referenced by: frec2uzf1od 10349 nnti 13949 |
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