Mathbox for BJ < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >   Mathboxes  >  bj-bdfindis Unicode version

Theorem bj-bdfindis 13336
 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 4523 for a proof of full induction in IZF. From this version, it is easy to prove bounded versions of finds 4523, finds2 4524, finds1 4525. (Contributed by BJ, 21-Nov-2019.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
bj-bdfindis.bd BOUNDED
bj-bdfindis.nf0
bj-bdfindis.nf1
bj-bdfindis.nfsuc
bj-bdfindis.0
bj-bdfindis.1
bj-bdfindis.suc
Assertion
Ref Expression
bj-bdfindis
Distinct variable groups:   ,   ,
Allowed substitution hints:   ()   (,)   (,)   (,)

Proof of Theorem bj-bdfindis
StepHypRef Expression
1 bj-bdfindis.nf0 . . . 4
2 0ex 4064 . . . 4
3 bj-bdfindis.0 . . . 4
41, 2, 3elabf2 13180 . . 3
5 bj-bdfindis.nf1 . . . . . 6
6 bj-bdfindis.1 . . . . . 6
75, 6elabf1 13179 . . . . 5
8 bj-bdfindis.nfsuc . . . . . 6
9 vex 2693 . . . . . . 7
109bj-sucex 13312 . . . . . 6
11 bj-bdfindis.suc . . . . . 6
128, 10, 11elabf2 13180 . . . . 5
137, 12imim12i 59 . . . 4
1413ralimi 2499 . . 3
15 bj-bdfindis.bd . . . . 5 BOUNDED
1615bdcab 13238 . . . 4 BOUNDED
1716bdpeano5 13332 . . 3
184, 14, 17syl2an 287 . 2
19 ssabral 3174 . 2
2018, 19sylib 121 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 103   wceq 1332  wnf 1437   wcel 1481  cab 2126  wral 2417   wss 3077  c0 3369   csuc 4296  com 4513  BOUNDED wbd 13201 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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-nul 4063  ax-pr 4140  ax-un 4364  ax-bd0 13202  ax-bdor 13205  ax-bdex 13208  ax-bdeq 13209  ax-bdel 13210  ax-bdsb 13211  ax-bdsep 13273  ax-infvn 13330 This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2692  df-dif 3079  df-un 3081  df-in 3083  df-ss 3090  df-nul 3370  df-sn 3539  df-pr 3540  df-uni 3746  df-int 3781  df-suc 4302  df-iom 4514  df-bdc 13230  df-bj-ind 13316 This theorem is referenced by:  bj-bdfindisg  13337  bj-bdfindes  13338  bj-nn0suc0  13339
 Copyright terms: Public domain W3C validator