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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 7270 | . . 3 | |
2 | 1 | simprbi 273 | . 2 |
3 | noel 3418 | . . . . 5 | |
4 | 1pi 7277 | . . . . . . . . 9 | |
5 | ltpiord 7281 | . . . . . . . . 9 | |
6 | 4, 5 | mpan2 423 | . . . . . . . 8 |
7 | df-1o 6395 | . . . . . . . . . 10 | |
8 | 7 | eleq2i 2237 | . . . . . . . . 9 |
9 | elsucg 4389 | . . . . . . . . 9 | |
10 | 8, 9 | syl5bb 191 | . . . . . . . 8 |
11 | 6, 10 | bitrd 187 | . . . . . . 7 |
12 | 11 | biimpa 294 | . . . . . 6 |
13 | 12 | ord 719 | . . . . 5 |
14 | 3, 13 | mpi 15 | . . . 4 |
15 | 14 | ex 114 | . . 3 |
16 | 15 | necon3ad 2382 | . 2 |
17 | 2, 16 | mpd 13 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wo 703 wceq 1348 wcel 2141 wne 2340 c0 3414 class class class wbr 3989 csuc 4350 com 4574 c1o 6388 cnpi 7234 clti 7237 |
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 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 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 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-br 3990 df-opab 4051 df-eprel 4274 df-suc 4356 df-iom 4575 df-xp 4617 df-1o 6395 df-ni 7266 df-lti 7269 |
This theorem is referenced by: caucvgsr 7764 |
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