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Mirrors > Home > ILE Home > Th. List > omsinds | Unicode version |
Description: Strong (or "total") induction principle over . (Contributed by Scott Fenton, 17-Jul-2015.) |
Ref | Expression |
---|---|
omsinds.1 | |
omsinds.2 | |
omsinds.3 |
Ref | Expression |
---|---|
omsinds |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | omsinds.2 | . 2 | |
2 | suceq 4324 | . . . 4 | |
3 | 2 | raleqdv 2632 | . . 3 |
4 | suceq 4324 | . . . 4 | |
5 | 4 | raleqdv 2632 | . . 3 |
6 | suceq 4324 | . . . 4 | |
7 | 6 | raleqdv 2632 | . . 3 |
8 | suceq 4324 | . . . 4 | |
9 | 8 | raleqdv 2632 | . . 3 |
10 | ral0 3464 | . . . . . 6 | |
11 | omsinds.3 | . . . . . . . 8 | |
12 | 11 | rgen 2485 | . . . . . . 7 |
13 | peano1 4508 | . . . . . . . 8 | |
14 | 10 | nfth 1440 | . . . . . . . . . 10 |
15 | nfsbc1v 2927 | . . . . . . . . . 10 | |
16 | 14, 15 | nfim 1551 | . . . . . . . . 9 |
17 | raleq 2626 | . . . . . . . . . 10 | |
18 | sbceq1a 2918 | . . . . . . . . . 10 | |
19 | 17, 18 | imbi12d 233 | . . . . . . . . 9 |
20 | 16, 19 | rspc 2783 | . . . . . . . 8 |
21 | 13, 20 | ax-mp 5 | . . . . . . 7 |
22 | 12, 21 | ax-mp 5 | . . . . . 6 |
23 | 10, 22 | ax-mp 5 | . . . . 5 |
24 | ralsns 3562 | . . . . . 6 | |
25 | 13, 24 | ax-mp 5 | . . . . 5 |
26 | 23, 25 | mpbir 145 | . . . 4 |
27 | suc0 4333 | . . . . 5 | |
28 | 27 | raleqi 2630 | . . . 4 |
29 | 26, 28 | mpbir 145 | . . 3 |
30 | simpr 109 | . . . . . 6 | |
31 | peano2 4509 | . . . . . . . . 9 | |
32 | 31 | adantr 274 | . . . . . . . 8 |
33 | omsinds.1 | . . . . . . . . . 10 | |
34 | 33 | cbvralv 2654 | . . . . . . . . 9 |
35 | 30, 34 | sylib 121 | . . . . . . . 8 |
36 | nfv 1508 | . . . . . . . . . . 11 | |
37 | nfsbc1v 2927 | . . . . . . . . . . 11 | |
38 | 36, 37 | nfim 1551 | . . . . . . . . . 10 |
39 | raleq 2626 | . . . . . . . . . . 11 | |
40 | sbceq1a 2918 | . . . . . . . . . . 11 | |
41 | 39, 40 | imbi12d 233 | . . . . . . . . . 10 |
42 | 38, 41 | rspc 2783 | . . . . . . . . 9 |
43 | 12, 42 | mpi 15 | . . . . . . . 8 |
44 | 32, 35, 43 | sylc 62 | . . . . . . 7 |
45 | ralsns 3562 | . . . . . . . 8 | |
46 | 32, 45 | syl 14 | . . . . . . 7 |
47 | 44, 46 | mpbird 166 | . . . . . 6 |
48 | ralun 3258 | . . . . . 6 | |
49 | 30, 47, 48 | syl2anc 408 | . . . . 5 |
50 | df-suc 4293 | . . . . . . 7 | |
51 | 50 | a1i 9 | . . . . . 6 |
52 | 51 | raleqdv 2632 | . . . . 5 |
53 | 49, 52 | mpbird 166 | . . . 4 |
54 | 53 | ex 114 | . . 3 |
55 | 3, 5, 7, 9, 29, 54 | finds 4514 | . 2 |
56 | sucidg 4338 | . 2 | |
57 | 1, 55, 56 | rspcdva 2794 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1331 wcel 1480 wral 2416 wsbc 2909 cun 3069 c0 3363 csn 3527 csuc 4287 com 4504 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-iinf 4502 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-uni 3737 df-int 3772 df-suc 4293 df-iom 4505 |
This theorem is referenced by: nninfalllem1 13203 nninfsellemqall 13211 |
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