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Mirrors > Home > ILE Home > Th. List > nn0o1gt2 | Unicode version |
Description: An odd nonnegative integer is either 1 or greater than 2. (Contributed by AV, 2-Jun-2020.) |
Ref | Expression |
---|---|
nn0o1gt2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elnn0 8972 | . . 3 | |
2 | elnnnn0c 9015 | . . . . 5 | |
3 | 1z 9073 | . . . . . . . 8 | |
4 | nn0z 9067 | . . . . . . . 8 | |
5 | zleloe 9094 | . . . . . . . 8 | |
6 | 3, 4, 5 | sylancr 410 | . . . . . . 7 |
7 | 1zzd 9074 | . . . . . . . . . . . . 13 | |
8 | zltp1le 9101 | . . . . . . . . . . . . 13 | |
9 | 7, 4, 8 | syl2anc 408 | . . . . . . . . . . . 12 |
10 | 1p1e2 8830 | . . . . . . . . . . . . . 14 | |
11 | 10 | breq1i 3931 | . . . . . . . . . . . . 13 |
12 | 11 | a1i 9 | . . . . . . . . . . . 12 |
13 | 2z 9075 | . . . . . . . . . . . . 13 | |
14 | zleloe 9094 | . . . . . . . . . . . . 13 | |
15 | 13, 4, 14 | sylancr 410 | . . . . . . . . . . . 12 |
16 | 9, 12, 15 | 3bitrd 213 | . . . . . . . . . . 11 |
17 | olc 700 | . . . . . . . . . . . . . 14 | |
18 | 17 | 2a1d 23 | . . . . . . . . . . . . 13 |
19 | oveq1 5774 | . . . . . . . . . . . . . . . . . . . 20 | |
20 | 19 | oveq1d 5782 | . . . . . . . . . . . . . . . . . . 19 |
21 | 20 | eqcoms 2140 | . . . . . . . . . . . . . . . . . 18 |
22 | 21 | adantl 275 | . . . . . . . . . . . . . . . . 17 |
23 | 2p1e3 8846 | . . . . . . . . . . . . . . . . . 18 | |
24 | 23 | oveq1i 5777 | . . . . . . . . . . . . . . . . 17 |
25 | 22, 24 | syl6eq 2186 | . . . . . . . . . . . . . . . 16 |
26 | 25 | eleq1d 2206 | . . . . . . . . . . . . . . 15 |
27 | 3halfnz 9141 | . . . . . . . . . . . . . . . 16 | |
28 | nn0z 9067 | . . . . . . . . . . . . . . . . 17 | |
29 | 28 | pm2.24d 611 | . . . . . . . . . . . . . . . 16 |
30 | 27, 29 | mpi 15 | . . . . . . . . . . . . . . 15 |
31 | 26, 30 | syl6bi 162 | . . . . . . . . . . . . . 14 |
32 | 31 | expcom 115 | . . . . . . . . . . . . 13 |
33 | 18, 32 | jaoi 705 | . . . . . . . . . . . 12 |
34 | 33 | com12 30 | . . . . . . . . . . 11 |
35 | 16, 34 | sylbid 149 | . . . . . . . . . 10 |
36 | 35 | com12 30 | . . . . . . . . 9 |
37 | orc 701 | . . . . . . . . . . 11 | |
38 | 37 | eqcoms 2140 | . . . . . . . . . 10 |
39 | 38 | 2a1d 23 | . . . . . . . . 9 |
40 | 36, 39 | jaoi 705 | . . . . . . . 8 |
41 | 40 | com12 30 | . . . . . . 7 |
42 | 6, 41 | sylbid 149 | . . . . . 6 |
43 | 42 | imp 123 | . . . . 5 |
44 | 2, 43 | sylbi 120 | . . . 4 |
45 | oveq1 5774 | . . . . . . . 8 | |
46 | 0p1e1 8827 | . . . . . . . 8 | |
47 | 45, 46 | syl6eq 2186 | . . . . . . 7 |
48 | 47 | oveq1d 5782 | . . . . . 6 |
49 | 48 | eleq1d 2206 | . . . . 5 |
50 | halfnz 9140 | . . . . . 6 | |
51 | nn0z 9067 | . . . . . . 7 | |
52 | 51 | pm2.24d 611 | . . . . . 6 |
53 | 50, 52 | mpi 15 | . . . . 5 |
54 | 49, 53 | syl6bi 162 | . . . 4 |
55 | 44, 54 | jaoi 705 | . . 3 |
56 | 1, 55 | sylbi 120 | . 2 |
57 | 56 | imp 123 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wo 697 wceq 1331 wcel 1480 class class class wbr 3924 (class class class)co 5767 cc0 7613 c1 7614 caddc 7616 clt 7793 cle 7794 cdiv 8425 cn 8713 c2 8764 c3 8765 cn0 8970 cz 9047 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-cnex 7704 ax-resscn 7705 ax-1cn 7706 ax-1re 7707 ax-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-mulrcl 7712 ax-addcom 7713 ax-mulcom 7714 ax-addass 7715 ax-mulass 7716 ax-distr 7717 ax-i2m1 7718 ax-0lt1 7719 ax-1rid 7720 ax-0id 7721 ax-rnegex 7722 ax-precex 7723 ax-cnre 7724 ax-pre-ltirr 7725 ax-pre-ltwlin 7726 ax-pre-lttrn 7727 ax-pre-apti 7728 ax-pre-ltadd 7729 ax-pre-mulgt0 7730 ax-pre-mulext 7731 |
This theorem depends on definitions: df-bi 116 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-reu 2421 df-rmo 2422 df-rab 2423 df-v 2683 df-sbc 2905 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-br 3925 df-opab 3985 df-id 4210 df-po 4213 df-iso 4214 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-iota 5083 df-fun 5120 df-fv 5126 df-riota 5723 df-ov 5770 df-oprab 5771 df-mpo 5772 df-pnf 7795 df-mnf 7796 df-xr 7797 df-ltxr 7798 df-le 7799 df-sub 7928 df-neg 7929 df-reap 8330 df-ap 8337 df-div 8426 df-inn 8714 df-2 8772 df-3 8773 df-4 8774 df-n0 8971 df-z 9048 |
This theorem is referenced by: nno 11592 nn0o 11593 |
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