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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 7114 | . . . . . 6 | |
3 | nn1suc.1 | . . . . . 6 | |
4 | 2, 3 | sbcie 2848 | . . . . 5 |
5 | 1, 4 | mpbir 144 | . . . 4 |
6 | 1nn 8050 | . . . . . . 7 | |
7 | eleq1 2141 | . . . . . . 7 | |
8 | 6, 7 | mpbiri 166 | . . . . . 6 |
9 | nn1suc.4 | . . . . . . 7 | |
10 | 9 | sbcieg 2846 | . . . . . 6 |
11 | 8, 10 | syl 14 | . . . . 5 |
12 | dfsbcq 2817 | . . . . 5 | |
13 | 11, 12 | bitr3d 188 | . . . 4 |
14 | 5, 13 | mpbiri 166 | . . 3 |
15 | 14 | a1i 9 | . 2 |
16 | elisset 2613 | . . . 4 | |
17 | eleq1 2141 | . . . . . 6 | |
18 | 17 | pm5.32ri 442 | . . . . 5 |
19 | nn1suc.6 | . . . . . . 7 | |
20 | 19 | adantr 270 | . . . . . 6 |
21 | nnre 8046 | . . . . . . . . 9 | |
22 | peano2re 7244 | . . . . . . . . 9 | |
23 | nn1suc.3 | . . . . . . . . . 10 | |
24 | 23 | sbcieg 2846 | . . . . . . . . 9 |
25 | 21, 22, 24 | 3syl 17 | . . . . . . . 8 |
26 | 25 | adantr 270 | . . . . . . 7 |
27 | oveq1 5539 | . . . . . . . . 9 | |
28 | 27 | sbceq1d 2820 | . . . . . . . 8 |
29 | 28 | adantl 271 | . . . . . . 7 |
30 | 26, 29 | bitr3d 188 | . . . . . 6 |
31 | 20, 30 | mpbid 145 | . . . . 5 |
32 | 18, 31 | sylbir 133 | . . . 4 |
33 | 16, 32 | exlimddv 1819 | . . 3 |
34 | nncn 8047 | . . . . . 6 | |
35 | ax-1cn 7069 | . . . . . 6 | |
36 | npcan 7317 | . . . . . 6 | |
37 | 34, 35, 36 | sylancl 404 | . . . . 5 |
38 | 37 | sbceq1d 2820 | . . . 4 |
39 | 38, 10 | bitrd 186 | . . 3 |
40 | 33, 39 | syl5ib 152 | . 2 |
41 | nn1m1nn 8057 | . 2 | |
42 | 15, 40, 41 | mpjaod 670 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wb 103 wceq 1284 wcel 1433 wsbc 2815 (class class class)co 5532 cc 6979 cr 6980 c1 6982 caddc 6984 cmin 7279 cn 8039 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-setind 4280 ax-cnex 7067 ax-resscn 7068 ax-1cn 7069 ax-1re 7070 ax-icn 7071 ax-addcl 7072 ax-addrcl 7073 ax-mulcl 7074 ax-addcom 7076 ax-addass 7078 ax-distr 7080 ax-i2m1 7081 ax-0id 7084 ax-rnegex 7085 ax-cnre 7087 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-br 3786 df-opab 3840 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-iota 4887 df-fun 4924 df-fv 4930 df-riota 5488 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-sub 7281 df-inn 8040 |
This theorem is referenced by: (None) |
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