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Mirrors > Home > ILE Home > Th. List > peano5nni | Unicode version |
Description: Peano's inductive postulate. Theorem I.36 (principle of mathematical induction) of [Apostol] p. 34. (Contributed by NM, 10-Jan-1997.) (Revised by Mario Carneiro, 17-Nov-2014.) |
Ref | Expression |
---|---|
peano5nni |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1re 7026 | . . . 4 | |
2 | elin 3126 | . . . . 5 | |
3 | 2 | biimpri 124 | . . . 4 |
4 | 1, 3 | mpan2 401 | . . 3 |
5 | inss1 3157 | . . . . 5 | |
6 | ssralv 3004 | . . . . 5 | |
7 | 5, 6 | ax-mp 7 | . . . 4 |
8 | inss2 3158 | . . . . . . . 8 | |
9 | 8 | sseli 2941 | . . . . . . 7 |
10 | 1red 7042 | . . . . . . 7 | |
11 | 9, 10 | readdcld 7055 | . . . . . 6 |
12 | elin 3126 | . . . . . . 7 | |
13 | 12 | simplbi2com 1333 | . . . . . 6 |
14 | 11, 13 | syl 14 | . . . . 5 |
15 | 14 | ralimia 2382 | . . . 4 |
16 | 7, 15 | syl 14 | . . 3 |
17 | reex 7015 | . . . . 5 | |
18 | 17 | inex2 3892 | . . . 4 |
19 | eleq2 2101 | . . . . . . 7 | |
20 | eleq2 2101 | . . . . . . . 8 | |
21 | 20 | raleqbi1dv 2513 | . . . . . . 7 |
22 | 19, 21 | anbi12d 442 | . . . . . 6 |
23 | 22 | elabg 2688 | . . . . 5 |
24 | dfnn2 7916 | . . . . . 6 | |
25 | intss1 3630 | . . . . . 6 | |
26 | 24, 25 | syl5eqss 2989 | . . . . 5 |
27 | 23, 26 | syl6bir 153 | . . . 4 |
28 | 18, 27 | ax-mp 7 | . . 3 |
29 | 4, 16, 28 | syl2an 273 | . 2 |
30 | 29, 5 | syl6ss 2957 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wceq 1243 wcel 1393 cab 2026 wral 2306 cvv 2557 cin 2916 wss 2917 cint 3615 (class class class)co 5512 cr 6888 c1 6890 caddc 6892 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-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-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-cnex 6975 ax-resscn 6976 ax-1re 6978 ax-addrcl 6981 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-v 2559 df-in 2924 df-ss 2931 df-int 3616 df-inn 7915 |
This theorem is referenced by: nnssre 7918 nnind 7930 |
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