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Theorem bdfind 13203
Description: Bounded induction (principle of induction when  A is assumed to be bounded), proved from basic constructive axioms. See find 4513 for a nonconstructive proof of the general case. See findset 13202 for a proof when  A is assumed to be a set. (Contributed by BJ, 22-Nov-2019.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
bdfind.bd  |- BOUNDED  A
Assertion
Ref Expression
bdfind  |-  ( ( A  C_  om  /\  (/)  e.  A  /\  A. x  e.  A  suc  x  e.  A )  ->  A  =  om )
Distinct variable group:    x, A

Proof of Theorem bdfind
StepHypRef Expression
1 bdfind.bd . . . 4  |- BOUNDED  A
2 bj-omex 13199 . . . 4  |-  om  e.  _V
31, 2bdssex 13159 . . 3  |-  ( A 
C_  om  ->  A  e. 
_V )
433ad2ant1 1002 . 2  |-  ( ( A  C_  om  /\  (/)  e.  A  /\  A. x  e.  A  suc  x  e.  A )  ->  A  e.  _V )
5 findset 13202 . 2  |-  ( A  e.  _V  ->  (
( A  C_  om  /\  (/) 
e.  A  /\  A. x  e.  A  suc  x  e.  A )  ->  A  =  om )
)
64, 5mpcom 36 1  |-  ( ( A  C_  om  /\  (/)  e.  A  /\  A. x  e.  A  suc  x  e.  A )  ->  A  =  om )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 962    = wceq 1331    e. wcel 1480   A.wral 2416   _Vcvv 2686    C_ wss 3071   (/)c0 3363   suc csuc 4287   omcom 4504  BOUNDED wbdc 13097
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 13070  ax-bdan 13072  ax-bdor 13073  ax-bdex 13076  ax-bdeq 13077  ax-bdel 13078  ax-bdsb 13079  ax-bdsep 13141  ax-infvn 13198
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 13098  df-bj-ind 13184
This theorem is referenced by: (None)
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