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Theorem bdfind 15095
Description: Bounded induction (principle of induction when 𝐴 is assumed to be bounded), proved from basic constructive axioms. See find 4613 for a nonconstructive proof of the general case. See findset 15094 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 ⊆ ω ∧ ∅ ∈ 𝐴 ∧ ∀𝑥𝐴 suc 𝑥𝐴) → 𝐴 = ω)
Distinct variable group:   𝑥,𝐴

Proof of Theorem bdfind
StepHypRef Expression
1 bdfind.bd . . . 4 BOUNDED 𝐴
2 bj-omex 15091 . . . 4 ω ∈ V
31, 2bdssex 15051 . . 3 (𝐴 ⊆ ω → 𝐴 ∈ V)
433ad2ant1 1020 . 2 ((𝐴 ⊆ ω ∧ ∅ ∈ 𝐴 ∧ ∀𝑥𝐴 suc 𝑥𝐴) → 𝐴 ∈ V)
5 findset 15094 . 2 (𝐴 ∈ V → ((𝐴 ⊆ ω ∧ ∅ ∈ 𝐴 ∧ ∀𝑥𝐴 suc 𝑥𝐴) → 𝐴 = ω))
64, 5mpcom 36 1 ((𝐴 ⊆ ω ∧ ∅ ∈ 𝐴 ∧ ∀𝑥𝐴 suc 𝑥𝐴) → 𝐴 = ω)
Colors of variables: wff set class
Syntax hints:  wi 4  w3a 980   = wceq 1364  wcel 2160  wral 2468  Vcvv 2752  wss 3144  c0 3437  suc csuc 4380  ωcom 4604  BOUNDED wbdc 14989
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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-nul 4144  ax-pr 4224  ax-un 4448  ax-bd0 14962  ax-bdan 14964  ax-bdor 14965  ax-bdex 14968  ax-bdeq 14969  ax-bdel 14970  ax-bdsb 14971  ax-bdsep 15033  ax-infvn 15090
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-sn 3613  df-pr 3614  df-uni 3825  df-int 3860  df-suc 4386  df-iom 4605  df-bdc 14990  df-bj-ind 15076
This theorem is referenced by: (None)
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