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Mirrors > Home > ILE Home > Th. List > nneoor | Unicode version |
Description: A positive integer is even or odd. (Contributed by Jim Kingdon, 15-Mar-2020.) |
Ref | Expression |
---|---|
nneoor |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq1 5872 | . . . . . 6 | |
2 | 1 | oveq1d 5880 | . . . . 5 |
3 | 2 | eleq1d 2244 | . . . 4 |
4 | oveq1 5872 | . . . . 5 | |
5 | 4 | eleq1d 2244 | . . . 4 |
6 | 3, 5 | orbi12d 793 | . . 3 |
7 | oveq1 5872 | . . . . . 6 | |
8 | 7 | oveq1d 5880 | . . . . 5 |
9 | 8 | eleq1d 2244 | . . . 4 |
10 | oveq1 5872 | . . . . 5 | |
11 | 10 | eleq1d 2244 | . . . 4 |
12 | 9, 11 | orbi12d 793 | . . 3 |
13 | oveq1 5872 | . . . . . 6 | |
14 | 13 | oveq1d 5880 | . . . . 5 |
15 | 14 | eleq1d 2244 | . . . 4 |
16 | oveq1 5872 | . . . . 5 | |
17 | 16 | eleq1d 2244 | . . . 4 |
18 | 15, 17 | orbi12d 793 | . . 3 |
19 | oveq1 5872 | . . . . . 6 | |
20 | 19 | oveq1d 5880 | . . . . 5 |
21 | 20 | eleq1d 2244 | . . . 4 |
22 | oveq1 5872 | . . . . 5 | |
23 | 22 | eleq1d 2244 | . . . 4 |
24 | 21, 23 | orbi12d 793 | . . 3 |
25 | df-2 8949 | . . . . . . 7 | |
26 | 25 | oveq1i 5875 | . . . . . 6 |
27 | 2div2e1 9022 | . . . . . 6 | |
28 | 26, 27 | eqtr3i 2198 | . . . . 5 |
29 | 1nn 8901 | . . . . 5 | |
30 | 28, 29 | eqeltri 2248 | . . . 4 |
31 | 30 | orci 731 | . . 3 |
32 | peano2nn 8902 | . . . . . 6 | |
33 | nncn 8898 | . . . . . . . 8 | |
34 | add1p1 9139 | . . . . . . . . . 10 | |
35 | 34 | oveq1d 5880 | . . . . . . . . 9 |
36 | 2cn 8961 | . . . . . . . . . . 11 | |
37 | 2ap0 8983 | . . . . . . . . . . . 12 # | |
38 | 36, 37 | pm3.2i 272 | . . . . . . . . . . 11 # |
39 | divdirap 8626 | . . . . . . . . . . 11 # | |
40 | 36, 38, 39 | mp3an23 1329 | . . . . . . . . . 10 |
41 | 27 | oveq2i 5876 | . . . . . . . . . 10 |
42 | 40, 41 | eqtrdi 2224 | . . . . . . . . 9 |
43 | 35, 42 | eqtrd 2208 | . . . . . . . 8 |
44 | 33, 43 | syl 14 | . . . . . . 7 |
45 | 44 | eleq1d 2244 | . . . . . 6 |
46 | 32, 45 | syl5ibr 156 | . . . . 5 |
47 | 46 | orim2d 788 | . . . 4 |
48 | orcom 728 | . . . 4 | |
49 | 47, 48 | syl6ib 161 | . . 3 |
50 | 6, 12, 18, 24, 31, 49 | nnind 8906 | . 2 |
51 | 50 | orcomd 729 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 104 wo 708 wceq 1353 wcel 2146 class class class wbr 3998 (class class class)co 5865 cc 7784 cc0 7786 c1 7787 caddc 7789 # cap 8512 cdiv 8601 cn 8890 c2 8941 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1445 ax-7 1446 ax-gen 1447 ax-ie1 1491 ax-ie2 1492 ax-8 1502 ax-10 1503 ax-11 1504 ax-i12 1505 ax-bndl 1507 ax-4 1508 ax-17 1524 ax-i9 1528 ax-ial 1532 ax-i5r 1533 ax-13 2148 ax-14 2149 ax-ext 2157 ax-sep 4116 ax-pow 4169 ax-pr 4203 ax-un 4427 ax-setind 4530 ax-cnex 7877 ax-resscn 7878 ax-1cn 7879 ax-1re 7880 ax-icn 7881 ax-addcl 7882 ax-addrcl 7883 ax-mulcl 7884 ax-mulrcl 7885 ax-addcom 7886 ax-mulcom 7887 ax-addass 7888 ax-mulass 7889 ax-distr 7890 ax-i2m1 7891 ax-0lt1 7892 ax-1rid 7893 ax-0id 7894 ax-rnegex 7895 ax-precex 7896 ax-cnre 7897 ax-pre-ltirr 7898 ax-pre-ltwlin 7899 ax-pre-lttrn 7900 ax-pre-apti 7901 ax-pre-ltadd 7902 ax-pre-mulgt0 7903 ax-pre-mulext 7904 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1459 df-sb 1761 df-eu 2027 df-mo 2028 df-clab 2162 df-cleq 2168 df-clel 2171 df-nfc 2306 df-ne 2346 df-nel 2441 df-ral 2458 df-rex 2459 df-reu 2460 df-rmo 2461 df-rab 2462 df-v 2737 df-sbc 2961 df-dif 3129 df-un 3131 df-in 3133 df-ss 3140 df-pw 3574 df-sn 3595 df-pr 3596 df-op 3598 df-uni 3806 df-int 3841 df-br 3999 df-opab 4060 df-id 4287 df-po 4290 df-iso 4291 df-xp 4626 df-rel 4627 df-cnv 4628 df-co 4629 df-dm 4630 df-iota 5170 df-fun 5210 df-fv 5216 df-riota 5821 df-ov 5868 df-oprab 5869 df-mpo 5870 df-pnf 7968 df-mnf 7969 df-xr 7970 df-ltxr 7971 df-le 7972 df-sub 8104 df-neg 8105 df-reap 8506 df-ap 8513 df-div 8602 df-inn 8891 df-2 8949 |
This theorem is referenced by: nneo 9327 zeo 9329 |
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