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Mirrors > Home > ILE Home > Th. List > nntr2 | Unicode version |
Description: Transitive law for natural numbers. (Contributed by Jim Kingdon, 22-Jul-2023.) |
Ref | Expression |
---|---|
nntr2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnon 4523 | . . . . 5 | |
2 | 1 | ad3antlr 484 | . . . 4 |
3 | simpr 109 | . . . . 5 | |
4 | simprr 521 | . . . . . 6 | |
5 | 4 | adantr 274 | . . . . 5 |
6 | 3, 5 | jca 304 | . . . 4 |
7 | ontr1 4311 | . . . 4 | |
8 | 2, 6, 7 | sylc 62 | . . 3 |
9 | simpr 109 | . . . 4 | |
10 | 4 | adantr 274 | . . . 4 |
11 | 9, 10 | eqeltrd 2216 | . . 3 |
12 | simplrl 524 | . . . . 5 | |
13 | simpr 109 | . . . . 5 | |
14 | 12, 13 | sseldd 3098 | . . . 4 |
15 | simplr 519 | . . . . . . 7 | |
16 | elnn 4519 | . . . . . . 7 | |
17 | 4, 15, 16 | syl2anc 408 | . . . . . 6 |
18 | 17 | adantr 274 | . . . . 5 |
19 | nnord 4525 | . . . . 5 | |
20 | ordirr 4457 | . . . . 5 | |
21 | 18, 19, 20 | 3syl 17 | . . . 4 |
22 | 14, 21 | pm2.21dd 609 | . . 3 |
23 | simpll 518 | . . . 4 | |
24 | nntri3or 6389 | . . . 4 | |
25 | 23, 17, 24 | syl2anc 408 | . . 3 |
26 | 8, 11, 22, 25 | mpjao3dan 1285 | . 2 |
27 | 26 | ex 114 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 w3o 961 wceq 1331 wcel 1480 wss 3071 word 4284 con0 4285 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: (None) |
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