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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 4433 | . . . . . 6 | |
2 | ltpiord 7095 | . . . . . . 7 | |
3 | 2 | anidms 394 | . . . . . 6 |
4 | 1, 3 | mtbiri 649 | . . . . 5 |
5 | 4 | adantl 275 | . . . 4 |
6 | pion 7086 | . . . . . . . 8 | |
7 | ontr1 4281 | . . . . . . . 8 | |
8 | 6, 7 | syl 14 | . . . . . . 7 |
9 | 8 | 3ad2ant3 989 | . . . . . 6 |
10 | ltpiord 7095 | . . . . . . . 8 | |
11 | 10 | 3adant3 986 | . . . . . . 7 |
12 | ltpiord 7095 | . . . . . . . 8 | |
13 | 12 | 3adant1 984 | . . . . . . 7 |
14 | 11, 13 | anbi12d 464 | . . . . . 6 |
15 | ltpiord 7095 | . . . . . . 7 | |
16 | 15 | 3adant2 985 | . . . . . 6 |
17 | 9, 14, 16 | 3imtr4d 202 | . . . . 5 |
18 | 17 | adantl 275 | . . . 4 |
19 | 5, 18 | ispod 4196 | . . 3 |
20 | pinn 7085 | . . . . . 6 | |
21 | pinn 7085 | . . . . . 6 | |
22 | nntri3or 6357 | . . . . . 6 | |
23 | 20, 21, 22 | syl2an 287 | . . . . 5 |
24 | biidd 171 | . . . . . 6 | |
25 | ltpiord 7095 | . . . . . . 7 | |
26 | 25 | ancoms 266 | . . . . . 6 |
27 | 10, 24, 26 | 3orbi123d 1274 | . . . . 5 |
28 | 23, 27 | mpbird 166 | . . . 4 |
29 | 28 | adantl 275 | . . 3 |
30 | 19, 29 | issod 4211 | . 2 |
31 | 30 | mptru 1325 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 w3o 946 w3a 947 wtru 1317 wcel 1465 class class class wbr 3899 wor 4187 con0 4255 com 4474 cnpi 7048 clti 7051 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-nul 4024 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-iinf 4472 |
This theorem depends on definitions: df-bi 116 df-3or 948 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-ral 2398 df-rex 2399 df-v 2662 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-br 3900 df-opab 3960 df-tr 3997 df-eprel 4181 df-po 4188 df-iso 4189 df-iord 4258 df-on 4260 df-suc 4263 df-iom 4475 df-xp 4515 df-ni 7080 df-lti 7083 |
This theorem is referenced by: ltsonq 7174 |
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