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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 4291 | . . . . . 6 | |
2 | ltpiord 6509 | . . . . . . 7 | |
3 | 2 | anidms 389 | . . . . . 6 |
4 | 1, 3 | mtbiri 632 | . . . . 5 |
5 | 4 | adantl 271 | . . . 4 |
6 | pion 6500 | . . . . . . . 8 | |
7 | ontr1 4144 | . . . . . . . 8 | |
8 | 6, 7 | syl 14 | . . . . . . 7 |
9 | 8 | 3ad2ant3 961 | . . . . . 6 |
10 | ltpiord 6509 | . . . . . . . 8 | |
11 | 10 | 3adant3 958 | . . . . . . 7 |
12 | ltpiord 6509 | . . . . . . . 8 | |
13 | 12 | 3adant1 956 | . . . . . . 7 |
14 | 11, 13 | anbi12d 456 | . . . . . 6 |
15 | ltpiord 6509 | . . . . . . 7 | |
16 | 15 | 3adant2 957 | . . . . . 6 |
17 | 9, 14, 16 | 3imtr4d 201 | . . . . 5 |
18 | 17 | adantl 271 | . . . 4 |
19 | 5, 18 | ispod 4059 | . . 3 |
20 | pinn 6499 | . . . . . 6 | |
21 | pinn 6499 | . . . . . 6 | |
22 | nntri3or 6095 | . . . . . 6 | |
23 | 20, 21, 22 | syl2an 283 | . . . . 5 |
24 | biidd 170 | . . . . . 6 | |
25 | ltpiord 6509 | . . . . . . 7 | |
26 | 25 | ancoms 264 | . . . . . 6 |
27 | 10, 24, 26 | 3orbi123d 1242 | . . . . 5 |
28 | 23, 27 | mpbird 165 | . . . 4 |
29 | 28 | adantl 271 | . . 3 |
30 | 19, 29 | issod 4074 | . 2 |
31 | 30 | trud 1293 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 102 wb 103 w3o 918 w3a 919 wtru 1285 wcel 1433 class class class wbr 3785 wor 4050 con0 4118 com 4331 cnpi 6462 clti 6465 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-nul 3904 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-iinf 4329 |
This theorem depends on definitions: df-bi 115 df-3or 920 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-ral 2353 df-rex 2354 df-v 2603 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-br 3786 df-opab 3840 df-tr 3876 df-eprel 4044 df-po 4051 df-iso 4052 df-iord 4121 df-on 4123 df-suc 4126 df-iom 4332 df-xp 4369 df-ni 6494 df-lti 6497 |
This theorem is referenced by: ltsonq 6588 |
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