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Mirrors > Home > ILE Home > Th. List > Mathboxes > bdfind | Unicode version |
Description: Bounded induction (principle of induction when is assumed to be bounded), proved from basic constructive axioms. See find 4513 for a nonconstructive proof of the general case. See findset 13143 for a proof when is assumed to be a set. (Contributed by BJ, 22-Nov-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bdfind.bd | BOUNDED |
Ref | Expression |
---|---|
bdfind |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bdfind.bd | . . . 4 BOUNDED | |
2 | bj-omex 13140 | . . . 4 | |
3 | 1, 2 | bdssex 13100 | . . 3 |
4 | 3 | 3ad2ant1 1002 | . 2 |
5 | findset 13143 | . 2 | |
6 | 4, 5 | mpcom 36 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 w3a 962 wceq 1331 wcel 1480 wral 2416 cvv 2686 wss 3071 c0 3363 csuc 4287 com 4504 BOUNDED wbdc 13038 |
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-nul 4054 ax-pr 4131 ax-un 4355 ax-bd0 13011 ax-bdan 13013 ax-bdor 13014 ax-bdex 13017 ax-bdeq 13018 ax-bdel 13019 ax-bdsb 13020 ax-bdsep 13082 ax-infvn 13139 |
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-rab 2425 df-v 2688 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-sn 3533 df-pr 3534 df-uni 3737 df-int 3772 df-suc 4293 df-iom 4505 df-bdc 13039 df-bj-ind 13125 |
This theorem is referenced by: (None) |
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