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Theorem bdfind 14320
Description: Bounded induction (principle of induction when  A is assumed to be bounded), proved from basic constructive axioms. See find 4594 for a nonconstructive proof of the general case. See findset 14319 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 14316 . . . 4  |-  om  e.  _V
31, 2bdssex 14276 . . 3  |-  ( A 
C_  om  ->  A  e. 
_V )
433ad2ant1 1018 . 2  |-  ( ( A  C_  om  /\  (/)  e.  A  /\  A. x  e.  A  suc  x  e.  A )  ->  A  e.  _V )
5 findset 14319 . 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 978    = wceq 1353    e. wcel 2148   A.wral 2455   _Vcvv 2737    C_ wss 3129   (/)c0 3422   suc csuc 4361   omcom 4585  BOUNDED wbdc 14214
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-nul 4126  ax-pr 4205  ax-un 4429  ax-bd0 14187  ax-bdan 14189  ax-bdor 14190  ax-bdex 14193  ax-bdeq 14194  ax-bdel 14195  ax-bdsb 14196  ax-bdsep 14258  ax-infvn 14315
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-sn 3597  df-pr 3598  df-uni 3808  df-int 3843  df-suc 4367  df-iom 4586  df-bdc 14215  df-bj-ind 14301
This theorem is referenced by: (None)
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