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Mirrors > Home > ILE Home > Th. List > nlt1pig | Unicode version |
Description: No positive integer is less than one. (Contributed by Jim Kingdon, 31-Aug-2019.) |
Ref | Expression |
---|---|
nlt1pig |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elni 7230 | . . 3 | |
2 | 1 | simprbi 273 | . 2 |
3 | noel 3399 | . . . . 5 | |
4 | 1pi 7237 | . . . . . . . . 9 | |
5 | ltpiord 7241 | . . . . . . . . 9 | |
6 | 4, 5 | mpan2 422 | . . . . . . . 8 |
7 | df-1o 6365 | . . . . . . . . . 10 | |
8 | 7 | eleq2i 2224 | . . . . . . . . 9 |
9 | elsucg 4366 | . . . . . . . . 9 | |
10 | 8, 9 | syl5bb 191 | . . . . . . . 8 |
11 | 6, 10 | bitrd 187 | . . . . . . 7 |
12 | 11 | biimpa 294 | . . . . . 6 |
13 | 12 | ord 714 | . . . . 5 |
14 | 3, 13 | mpi 15 | . . . 4 |
15 | 14 | ex 114 | . . 3 |
16 | 15 | necon3ad 2369 | . 2 |
17 | 2, 16 | mpd 13 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wo 698 wceq 1335 wcel 2128 wne 2327 c0 3395 class class class wbr 3967 csuc 4327 com 4551 c1o 6358 cnpi 7194 clti 7197 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4084 ax-nul 4092 ax-pow 4137 ax-pr 4171 ax-un 4395 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-ral 2440 df-rex 2441 df-v 2714 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3396 df-pw 3546 df-sn 3567 df-pr 3568 df-op 3570 df-uni 3775 df-int 3810 df-br 3968 df-opab 4028 df-eprel 4251 df-suc 4333 df-iom 4552 df-xp 4594 df-1o 6365 df-ni 7226 df-lti 7229 |
This theorem is referenced by: caucvgsr 7724 |
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