Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > find | Unicode version |
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 |
---|---|
find |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | find.1 | . . 3 | |
2 | 1 | simp1i 990 | . 2 |
3 | 3simpc 980 | . . . . 5 | |
4 | 1, 3 | ax-mp 5 | . . . 4 |
5 | df-ral 2421 | . . . . . 6 | |
6 | alral 2478 | . . . . . 6 | |
7 | 5, 6 | sylbi 120 | . . . . 5 |
8 | 7 | anim2i 339 | . . . 4 |
9 | 4, 8 | ax-mp 5 | . . 3 |
10 | peano5 4512 | . . 3 | |
11 | 9, 10 | ax-mp 5 | . 2 |
12 | 2, 11 | eqssi 3113 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 962 wal 1329 wceq 1331 wcel 1480 wral 2416 wss 3071 c0 3363 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-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: (None) |
Copyright terms: Public domain | W3C validator |