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Mirrors > Home > ILE Home > Th. List > ltapig | Unicode version |
Description: Ordering property of addition for positive integers. (Contributed by Jim Kingdon, 31-Aug-2019.) |
Ref | Expression |
---|---|
ltapig |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pinn 7271 | . . . . 5 | |
2 | pinn 7271 | . . . . 5 | |
3 | pinn 7271 | . . . . 5 | |
4 | nnaord 6488 | . . . . 5 | |
5 | 1, 2, 3, 4 | syl3an 1275 | . . . 4 |
6 | 5 | 3expa 1198 | . . 3 |
7 | ltpiord 7281 | . . . 4 | |
8 | 7 | adantr 274 | . . 3 |
9 | addclpi 7289 | . . . . . . 7 | |
10 | addclpi 7289 | . . . . . . 7 | |
11 | ltpiord 7281 | . . . . . . 7 | |
12 | 9, 10, 11 | syl2an 287 | . . . . . 6 |
13 | addpiord 7278 | . . . . . . . 8 | |
14 | 13 | adantr 274 | . . . . . . 7 |
15 | addpiord 7278 | . . . . . . . 8 | |
16 | 15 | adantl 275 | . . . . . . 7 |
17 | 14, 16 | eleq12d 2241 | . . . . . 6 |
18 | 12, 17 | bitrd 187 | . . . . 5 |
19 | 18 | anandis 587 | . . . 4 |
20 | 19 | ancoms 266 | . . 3 |
21 | 6, 8, 20 | 3bitr4d 219 | . 2 |
22 | 21 | 3impa 1189 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 973 wceq 1348 wcel 2141 class class class wbr 3989 com 4574 (class class class)co 5853 coa 6392 cnpi 7234 cpli 7235 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-coll 4104 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-iinf 4572 |
This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 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-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 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-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-tr 4088 df-eprel 4274 df-id 4278 df-iord 4351 df-on 4353 df-suc 4356 df-iom 4575 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-ov 5856 df-oprab 5857 df-mpo 5858 df-1st 6119 df-2nd 6120 df-recs 6284 df-irdg 6349 df-oadd 6399 df-ni 7266 df-pli 7267 df-lti 7269 |
This theorem is referenced by: ltanqg 7362 |
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