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Mirrors > Home > MPE Home > Th. List > nnneo | Structured version Visualization version Unicode version |
Description: If a natural number is even, its successor is odd. (Contributed by Mario Carneiro, 16-Nov-2014.) |
Ref | Expression |
---|---|
nnneo |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnon 7071 | . . . 4 | |
2 | onnbtwn 5818 | . . . 4 | |
3 | 1, 2 | syl 17 | . . 3 |
4 | 3 | 3ad2ant1 1082 | . 2 |
5 | suceq 5790 | . . . . 5 | |
6 | 5 | eqeq1d 2624 | . . . 4 |
7 | 6 | 3ad2ant3 1084 | . . 3 |
8 | ovex 6678 | . . . . . . . 8 | |
9 | 8 | sucid 5804 | . . . . . . 7 |
10 | eleq2 2690 | . . . . . . 7 | |
11 | 9, 10 | mpbii 223 | . . . . . 6 |
12 | 2onn 7720 | . . . . . . . 8 | |
13 | nnmord 7712 | . . . . . . . 8 | |
14 | 12, 13 | mp3an3 1413 | . . . . . . 7 |
15 | simpl 473 | . . . . . . 7 | |
16 | 14, 15 | syl6bir 244 | . . . . . 6 |
17 | 11, 16 | syl5 34 | . . . . 5 |
18 | simpr 477 | . . . . . . . . 9 | |
19 | nnmcl 7692 | . . . . . . . . . . . . 13 | |
20 | 12, 19 | mpan 706 | . . . . . . . . . . . 12 |
21 | nnon 7071 | . . . . . . . . . . . 12 | |
22 | oa1suc 7611 | . . . . . . . . . . . 12 | |
23 | 20, 21, 22 | 3syl 18 | . . . . . . . . . . 11 |
24 | 1onn 7719 | . . . . . . . . . . . . . . . 16 | |
25 | 24 | elexi 3213 | . . . . . . . . . . . . . . 15 |
26 | 25 | sucid 5804 | . . . . . . . . . . . . . 14 |
27 | df-2o 7561 | . . . . . . . . . . . . . 14 | |
28 | 26, 27 | eleqtrri 2700 | . . . . . . . . . . . . 13 |
29 | nnaord 7699 | . . . . . . . . . . . . . . 15 | |
30 | 24, 12, 29 | mp3an12 1414 | . . . . . . . . . . . . . 14 |
31 | 20, 30 | syl 17 | . . . . . . . . . . . . 13 |
32 | 28, 31 | mpbii 223 | . . . . . . . . . . . 12 |
33 | nnmsuc 7687 | . . . . . . . . . . . . 13 | |
34 | 12, 33 | mpan 706 | . . . . . . . . . . . 12 |
35 | 32, 34 | eleqtrrd 2704 | . . . . . . . . . . 11 |
36 | 23, 35 | eqeltrrd 2702 | . . . . . . . . . 10 |
37 | 36 | ad2antrr 762 | . . . . . . . . 9 |
38 | 18, 37 | eqeltrrd 2702 | . . . . . . . 8 |
39 | peano2 7086 | . . . . . . . . . . 11 | |
40 | nnmord 7712 | . . . . . . . . . . . 12 | |
41 | 12, 40 | mp3an3 1413 | . . . . . . . . . . 11 |
42 | 39, 41 | sylan2 491 | . . . . . . . . . 10 |
43 | 42 | ancoms 469 | . . . . . . . . 9 |
44 | 43 | adantr 481 | . . . . . . . 8 |
45 | 38, 44 | mpbird 247 | . . . . . . 7 |
46 | 45 | simpld 475 | . . . . . 6 |
47 | 46 | ex 450 | . . . . 5 |
48 | 17, 47 | jcad 555 | . . . 4 |
49 | 48 | 3adant3 1081 | . . 3 |
50 | 7, 49 | sylbid 230 | . 2 |
51 | 4, 50 | mtod 189 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wi 4 wb 196 wa 384 w3a 1037 wceq 1483 wcel 1990 c0 3915 con0 5723 csuc 5725 (class class class)co 6650 com 7065 c1o 7553 c2o 7554 coa 7557 comu 7558 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-omul 7565 |
This theorem is referenced by: nneob 7732 |
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