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Mirrors > Home > ILE Home > Th. List > nn1suc | Unicode version |
Description: If a statement holds for 1 and also holds for a successor, it holds for all positive integers. The first three hypotheses give us the substitution instances we need; the last two show that it holds for 1 and for a successor. (Contributed by NM, 11-Oct-2004.) (Revised by Mario Carneiro, 16-May-2014.) |
Ref | Expression |
---|---|
nn1suc.1 | |
nn1suc.3 | |
nn1suc.4 | |
nn1suc.5 | |
nn1suc.6 |
Ref | Expression |
---|---|
nn1suc |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn1suc.5 | . . . . 5 | |
2 | 1ex 7729 | . . . . . 6 | |
3 | nn1suc.1 | . . . . . 6 | |
4 | 2, 3 | sbcie 2915 | . . . . 5 |
5 | 1, 4 | mpbir 145 | . . . 4 |
6 | 1nn 8699 | . . . . . . 7 | |
7 | eleq1 2180 | . . . . . . 7 | |
8 | 6, 7 | mpbiri 167 | . . . . . 6 |
9 | nn1suc.4 | . . . . . . 7 | |
10 | 9 | sbcieg 2913 | . . . . . 6 |
11 | 8, 10 | syl 14 | . . . . 5 |
12 | dfsbcq 2884 | . . . . 5 | |
13 | 11, 12 | bitr3d 189 | . . . 4 |
14 | 5, 13 | mpbiri 167 | . . 3 |
15 | 14 | a1i 9 | . 2 |
16 | elisset 2674 | . . . 4 | |
17 | eleq1 2180 | . . . . . 6 | |
18 | 17 | pm5.32ri 450 | . . . . 5 |
19 | nn1suc.6 | . . . . . . 7 | |
20 | 19 | adantr 274 | . . . . . 6 |
21 | nnre 8695 | . . . . . . . . 9 | |
22 | peano2re 7866 | . . . . . . . . 9 | |
23 | nn1suc.3 | . . . . . . . . . 10 | |
24 | 23 | sbcieg 2913 | . . . . . . . . 9 |
25 | 21, 22, 24 | 3syl 17 | . . . . . . . 8 |
26 | 25 | adantr 274 | . . . . . . 7 |
27 | oveq1 5749 | . . . . . . . . 9 | |
28 | 27 | sbceq1d 2887 | . . . . . . . 8 |
29 | 28 | adantl 275 | . . . . . . 7 |
30 | 26, 29 | bitr3d 189 | . . . . . 6 |
31 | 20, 30 | mpbid 146 | . . . . 5 |
32 | 18, 31 | sylbir 134 | . . . 4 |
33 | 16, 32 | exlimddv 1854 | . . 3 |
34 | nncn 8696 | . . . . . 6 | |
35 | ax-1cn 7681 | . . . . . 6 | |
36 | npcan 7939 | . . . . . 6 | |
37 | 34, 35, 36 | sylancl 409 | . . . . 5 |
38 | 37 | sbceq1d 2887 | . . . 4 |
39 | 38, 10 | bitrd 187 | . . 3 |
40 | 33, 39 | syl5ib 153 | . 2 |
41 | nn1m1nn 8706 | . 2 | |
42 | 15, 40, 41 | mpjaod 692 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1316 wcel 1465 wsbc 2882 (class class class)co 5742 cc 7586 cr 7587 c1 7589 caddc 7591 cmin 7901 cn 8688 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-setind 4422 ax-cnex 7679 ax-resscn 7680 ax-1cn 7681 ax-1re 7682 ax-icn 7683 ax-addcl 7684 ax-addrcl 7685 ax-mulcl 7686 ax-addcom 7688 ax-addass 7690 ax-distr 7692 ax-i2m1 7693 ax-0id 7696 ax-rnegex 7697 ax-cnre 7699 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-ral 2398 df-rex 2399 df-reu 2400 df-rab 2402 df-v 2662 df-sbc 2883 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-br 3900 df-opab 3960 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-iota 5058 df-fun 5095 df-fv 5101 df-riota 5698 df-ov 5745 df-oprab 5746 df-mpo 5747 df-sub 7903 df-inn 8689 |
This theorem is referenced by: (None) |
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