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Mirrors > Home > ILE Home > Th. List > ltsopi | Unicode version |
Description: Positive integer 'less than' is a strict ordering. (Contributed by NM, 8-Feb-1996.) (Proof shortened by Mario Carneiro, 10-Jul-2014.) |
Ref | Expression |
---|---|
ltsopi |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elirrv 4272 | . . . . . 6 | |
2 | ltpiord 6417 | . . . . . . 7 | |
3 | 2 | anidms 377 | . . . . . 6 |
4 | 1, 3 | mtbiri 600 | . . . . 5 |
5 | 4 | adantl 262 | . . . 4 |
6 | pion 6408 | . . . . . . . 8 | |
7 | ontr1 4126 | . . . . . . . 8 | |
8 | 6, 7 | syl 14 | . . . . . . 7 |
9 | 8 | 3ad2ant3 927 | . . . . . 6 |
10 | ltpiord 6417 | . . . . . . . 8 | |
11 | 10 | 3adant3 924 | . . . . . . 7 |
12 | ltpiord 6417 | . . . . . . . 8 | |
13 | 12 | 3adant1 922 | . . . . . . 7 |
14 | 11, 13 | anbi12d 442 | . . . . . 6 |
15 | ltpiord 6417 | . . . . . . 7 | |
16 | 15 | 3adant2 923 | . . . . . 6 |
17 | 9, 14, 16 | 3imtr4d 192 | . . . . 5 |
18 | 17 | adantl 262 | . . . 4 |
19 | 5, 18 | ispod 4041 | . . 3 |
20 | pinn 6407 | . . . . . 6 | |
21 | pinn 6407 | . . . . . 6 | |
22 | nntri3or 6072 | . . . . . 6 | |
23 | 20, 21, 22 | syl2an 273 | . . . . 5 |
24 | biidd 161 | . . . . . 6 | |
25 | ltpiord 6417 | . . . . . . 7 | |
26 | 25 | ancoms 255 | . . . . . 6 |
27 | 10, 24, 26 | 3orbi123d 1206 | . . . . 5 |
28 | 23, 27 | mpbird 156 | . . . 4 |
29 | 28 | adantl 262 | . . 3 |
30 | 19, 29 | issod 4056 | . 2 |
31 | 30 | trud 1252 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 97 wb 98 w3o 884 w3a 885 wtru 1244 wcel 1393 class class class wbr 3764 wor 4032 con0 4100 com 4313 cnpi 6370 clti 6373 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-nul 3883 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 ax-iinf 4311 |
This theorem depends on definitions: df-bi 110 df-3or 886 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-ral 2311 df-rex 2312 df-v 2559 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-int 3616 df-br 3765 df-opab 3819 df-tr 3855 df-eprel 4026 df-po 4033 df-iso 4034 df-iord 4103 df-on 4105 df-suc 4108 df-iom 4314 df-xp 4351 df-ni 6402 df-lti 6405 |
This theorem is referenced by: ltsonq 6496 |
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