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Mirrors > Home > ILE Home > Th. List > zdclt | Unicode version |
Description: Integer is decidable. (Contributed by Jim Kingdon, 1-Jun-2020.) |
Ref | Expression |
---|---|
zdclt | DECID |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ztri3or 9205 | . 2 | |
2 | zre 9166 | . . 3 | |
3 | zre 9166 | . . 3 | |
4 | orc 702 | . . . . . 6 | |
5 | df-dc 821 | . . . . . 6 DECID | |
6 | 4, 5 | sylibr 133 | . . . . 5 DECID |
7 | 6 | a1i 9 | . . . 4 DECID |
8 | ltnr 7949 | . . . . . . . . 9 | |
9 | 8 | adantr 274 | . . . . . . . 8 |
10 | breq2 3969 | . . . . . . . . 9 | |
11 | 10 | adantl 275 | . . . . . . . 8 |
12 | 9, 11 | mtbid 662 | . . . . . . 7 |
13 | olc 701 | . . . . . . . 8 | |
14 | 13, 5 | sylibr 133 | . . . . . . 7 DECID |
15 | 12, 14 | syl 14 | . . . . . 6 DECID |
16 | 15 | ex 114 | . . . . 5 DECID |
17 | 16 | adantr 274 | . . . 4 DECID |
18 | ltnsym 7958 | . . . . . 6 | |
19 | 18 | ancoms 266 | . . . . 5 |
20 | 19, 14 | syl6 33 | . . . 4 DECID |
21 | 7, 17, 20 | 3jaod 1286 | . . 3 DECID |
22 | 2, 3, 21 | syl2an 287 | . 2 DECID |
23 | 1, 22 | mpd 13 | 1 DECID |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wo 698 DECID wdc 820 w3o 962 wceq 1335 wcel 2128 class class class wbr 3965 cr 7726 clt 7907 cz 9162 |
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 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4495 ax-cnex 7818 ax-resscn 7819 ax-1cn 7820 ax-1re 7821 ax-icn 7822 ax-addcl 7823 ax-addrcl 7824 ax-mulcl 7825 ax-addcom 7827 ax-addass 7829 ax-distr 7831 ax-i2m1 7832 ax-0lt1 7833 ax-0id 7835 ax-rnegex 7836 ax-cnre 7838 ax-pre-ltirr 7839 ax-pre-ltwlin 7840 ax-pre-lttrn 7841 ax-pre-ltadd 7843 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-reu 2442 df-rab 2444 df-v 2714 df-sbc 2938 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-int 3808 df-br 3966 df-opab 4026 df-id 4253 df-xp 4591 df-rel 4592 df-cnv 4593 df-co 4594 df-dm 4595 df-iota 5134 df-fun 5171 df-fv 5177 df-riota 5777 df-ov 5824 df-oprab 5825 df-mpo 5826 df-pnf 7909 df-mnf 7910 df-xr 7911 df-ltxr 7912 df-le 7913 df-sub 8043 df-neg 8044 df-inn 8829 df-n0 9086 df-z 9163 |
This theorem is referenced by: fztri3or 9936 modifeq2int 10280 modsumfzodifsn 10290 exp3val 10416 cvgratz 11424 |
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