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Description: The Principle of Finite Induction (mathematical induction). Corollary 7.31 of [TakeutiZaring] p. 43. The simpler hypothesis shown here was suggested in an email from "Colin" on 1-Oct-2001. The hypothesis states that is a set of natural numbers, zero belongs to , and given any member of the member's successor also belongs to . The conclusion is that every natural number is in . (Contributed by NM, 22-Feb-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Ref | Expression |
---|---|
find.1 |
Ref | Expression |
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find |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | find.1 | . . 3 | |
2 | 1 | simp1i 947 | . 2 |
3 | 3simpc 937 | . . . . 5 | |
4 | 1, 3 | ax-mp 7 | . . . 4 |
5 | df-ral 2353 | . . . . . 6 | |
6 | alral 2409 | . . . . . 6 | |
7 | 5, 6 | sylbi 119 | . . . . 5 |
8 | 7 | anim2i 334 | . . . 4 |
9 | 4, 8 | ax-mp 7 | . . 3 |
10 | peano5 4339 | . . 3 | |
11 | 9, 10 | ax-mp 7 | . 2 |
12 | 2, 11 | eqssi 3015 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 w3a 919 wal 1282 wceq 1284 wcel 1433 wral 2348 wss 2973 c0 3251 csuc 4120 com 4331 |
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-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-nul 3904 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-iinf 4329 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-pw 3384 df-sn 3404 df-pr 3405 df-uni 3602 df-int 3637 df-suc 4126 df-iom 4332 |
This theorem is referenced by: (None) |
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