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Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-bdfindis | Unicode version |
Description: Bounded induction (principle of induction for bounded formulas), using implicit substitutions (the biconditional versions of the hypotheses are implicit substitutions, and we have weakened them to implications). Constructive proof (from CZF). See finds 4571 for a proof of full induction in IZF. From this version, it is easy to prove bounded versions of finds 4571, finds2 4572, finds1 4573. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-bdfindis.bd | BOUNDED |
bj-bdfindis.nf0 | |
bj-bdfindis.nf1 | |
bj-bdfindis.nfsuc | |
bj-bdfindis.0 | |
bj-bdfindis.1 | |
bj-bdfindis.suc |
Ref | Expression |
---|---|
bj-bdfindis |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-bdfindis.nf0 | . . . 4 | |
2 | 0ex 4103 | . . . 4 | |
3 | bj-bdfindis.0 | . . . 4 | |
4 | 1, 2, 3 | elabf2 13504 | . . 3 |
5 | bj-bdfindis.nf1 | . . . . . 6 | |
6 | bj-bdfindis.1 | . . . . . 6 | |
7 | 5, 6 | elabf1 13503 | . . . . 5 |
8 | bj-bdfindis.nfsuc | . . . . . 6 | |
9 | vex 2724 | . . . . . . 7 | |
10 | 9 | bj-sucex 13646 | . . . . . 6 |
11 | bj-bdfindis.suc | . . . . . 6 | |
12 | 8, 10, 11 | elabf2 13504 | . . . . 5 |
13 | 7, 12 | imim12i 59 | . . . 4 |
14 | 13 | ralimi 2527 | . . 3 |
15 | bj-bdfindis.bd | . . . . 5 BOUNDED | |
16 | 15 | bdcab 13572 | . . . 4 BOUNDED |
17 | 16 | bdpeano5 13666 | . . 3 |
18 | 4, 14, 17 | syl2an 287 | . 2 |
19 | ssabral 3208 | . 2 | |
20 | 18, 19 | sylib 121 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1342 wnf 1447 wcel 2135 cab 2150 wral 2442 wss 3111 c0 3404 csuc 4337 com 4561 BOUNDED wbd 13535 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-nul 4102 ax-pr 4181 ax-un 4405 ax-bd0 13536 ax-bdor 13539 ax-bdex 13542 ax-bdeq 13543 ax-bdel 13544 ax-bdsb 13545 ax-bdsep 13607 ax-infvn 13664 |
This theorem depends on definitions: df-bi 116 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-rab 2451 df-v 2723 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-sn 3576 df-pr 3577 df-uni 3784 df-int 3819 df-suc 4343 df-iom 4562 df-bdc 13564 df-bj-ind 13650 |
This theorem is referenced by: bj-bdfindisg 13671 bj-bdfindes 13672 bj-nn0suc0 13673 |
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