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Mirrors > Home > ILE Home > Th. List > nnwetri | Unicode version |
Description: A natural number is well-ordered by . More specifically, this order both satisfies and is trichotomous. (Contributed by Jim Kingdon, 25-Sep-2021.) |
Ref | Expression |
---|---|
nnwetri |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnord 4583 | . . 3 | |
2 | ordwe 4547 | . . 3 | |
3 | 1, 2 | syl 14 | . 2 |
4 | simprl 521 | . . . . 5 | |
5 | simpl 108 | . . . . 5 | |
6 | elnn 4577 | . . . . 5 | |
7 | 4, 5, 6 | syl2anc 409 | . . . 4 |
8 | simprr 522 | . . . . 5 | |
9 | elnn 4577 | . . . . 5 | |
10 | 8, 5, 9 | syl2anc 409 | . . . 4 |
11 | nntri3or 6452 | . . . . 5 | |
12 | epel 4264 | . . . . . 6 | |
13 | biid 170 | . . . . . 6 | |
14 | epel 4264 | . . . . . 6 | |
15 | 12, 13, 14 | 3orbi123i 1178 | . . . . 5 |
16 | 11, 15 | sylibr 133 | . . . 4 |
17 | 7, 10, 16 | syl2anc 409 | . . 3 |
18 | 17 | ralrimivva 2546 | . 2 |
19 | 3, 18 | jca 304 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3o 966 wcel 2135 wral 2442 class class class wbr 3976 cep 4259 wwe 4302 word 4334 com 4561 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-iinf 4559 |
This theorem depends on definitions: df-bi 116 df-3or 968 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2723 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-br 3977 df-opab 4038 df-tr 4075 df-eprel 4261 df-frfor 4303 df-frind 4304 df-wetr 4306 df-iord 4338 df-on 4340 df-suc 4343 df-iom 4562 |
This theorem is referenced by: (None) |
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