Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 7022 | . . . . . 6 | |
3 | nn1suc.1 | . . . . . 6 | |
4 | 2, 3 | sbcie 2797 | . . . . 5 |
5 | 1, 4 | mpbir 134 | . . . 4 |
6 | 1nn 7925 | . . . . . . 7 | |
7 | eleq1 2100 | . . . . . . 7 | |
8 | 6, 7 | mpbiri 157 | . . . . . 6 |
9 | nn1suc.4 | . . . . . . 7 | |
10 | 9 | sbcieg 2795 | . . . . . 6 |
11 | 8, 10 | syl 14 | . . . . 5 |
12 | dfsbcq 2766 | . . . . 5 | |
13 | 11, 12 | bitr3d 179 | . . . 4 |
14 | 5, 13 | mpbiri 157 | . . 3 |
15 | 14 | a1i 9 | . 2 |
16 | elisset 2568 | . . . 4 | |
17 | eleq1 2100 | . . . . . 6 | |
18 | 17 | pm5.32ri 428 | . . . . 5 |
19 | nn1suc.6 | . . . . . . 7 | |
20 | 19 | adantr 261 | . . . . . 6 |
21 | nnre 7921 | . . . . . . . . 9 | |
22 | peano2re 7149 | . . . . . . . . 9 | |
23 | nn1suc.3 | . . . . . . . . . 10 | |
24 | 23 | sbcieg 2795 | . . . . . . . . 9 |
25 | 21, 22, 24 | 3syl 17 | . . . . . . . 8 |
26 | 25 | adantr 261 | . . . . . . 7 |
27 | oveq1 5519 | . . . . . . . . 9 | |
28 | 27 | sbceq1d 2769 | . . . . . . . 8 |
29 | 28 | adantl 262 | . . . . . . 7 |
30 | 26, 29 | bitr3d 179 | . . . . . 6 |
31 | 20, 30 | mpbid 135 | . . . . 5 |
32 | 18, 31 | sylbir 125 | . . . 4 |
33 | 16, 32 | exlimddv 1778 | . . 3 |
34 | nncn 7922 | . . . . . 6 | |
35 | ax-1cn 6977 | . . . . . 6 | |
36 | npcan 7220 | . . . . . 6 | |
37 | 34, 35, 36 | sylancl 392 | . . . . 5 |
38 | 37 | sbceq1d 2769 | . . . 4 |
39 | 38, 10 | bitrd 177 | . . 3 |
40 | 33, 39 | syl5ib 143 | . 2 |
41 | nn1m1nn 7932 | . 2 | |
42 | 15, 40, 41 | mpjaod 638 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wb 98 wceq 1243 wcel 1393 wsbc 2764 (class class class)co 5512 cc 6887 cr 6888 c1 6890 caddc 6892 cmin 7182 cn 7914 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 ax-setind 4262 ax-cnex 6975 ax-resscn 6976 ax-1cn 6977 ax-1re 6978 ax-icn 6979 ax-addcl 6980 ax-addrcl 6981 ax-mulcl 6982 ax-addcom 6984 ax-addass 6986 ax-distr 6988 ax-i2m1 6989 ax-0id 6992 ax-rnegex 6993 ax-cnre 6995 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-ral 2311 df-rex 2312 df-reu 2313 df-rab 2315 df-v 2559 df-sbc 2765 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-int 3616 df-br 3765 df-opab 3819 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-iota 4867 df-fun 4904 df-fv 4910 df-riota 5468 df-ov 5515 df-oprab 5516 df-mpt2 5517 df-sub 7184 df-inn 7915 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |