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 913 | . 2 |
3 | 3simpc 903 | . . . . 5 | |
4 | 1, 3 | ax-mp 7 | . . . 4 |
5 | df-ral 2311 | . . . . . 6 | |
6 | alral 2367 | . . . . . 6 | |
7 | 5, 6 | sylbi 114 | . . . . 5 |
8 | 7 | anim2i 324 | . . . 4 |
9 | 4, 8 | ax-mp 7 | . . 3 |
10 | peano5 4321 | . . 3 | |
11 | 9, 10 | ax-mp 7 | . 2 |
12 | 2, 11 | eqssi 2961 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 w3a 885 wal 1241 wceq 1243 wcel 1393 wral 2306 wss 2917 c0 3224 csuc 4102 com 4313 |
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-in1 544 ax-in2 545 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-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-nul 3883 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-iinf 4311 |
This theorem depends on definitions: df-bi 110 df-3an 887 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-rex 2312 df-v 2559 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-uni 3581 df-int 3616 df-suc 4108 df-iom 4314 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |