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Theorem bdfind 12565
Description: Bounded induction (principle of induction when 𝐴 is assumed to be bounded), proved from basic constructive axioms. See find 4442 for a nonconstructive proof of the general case. See findset 12564 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 12561 . . . 4 ω ∈ V
31, 2bdssex 12517 . . 3 (𝐴 ⊆ ω → 𝐴 ∈ V)
433ad2ant1 967 . 2 ((𝐴 ⊆ ω ∧ ∅ ∈ 𝐴 ∧ ∀𝑥𝐴 suc 𝑥𝐴) → 𝐴 ∈ V)
5 findset 12564 . 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 927   = wceq 1296  wcel 1445  wral 2370  Vcvv 2633  wss 3013  c0 3302  suc csuc 4216  ωcom 4433  BOUNDED wbdc 12455
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-nul 3986  ax-pr 4060  ax-un 4284  ax-bd0 12428  ax-bdan 12430  ax-bdor 12431  ax-bdex 12434  ax-bdeq 12435  ax-bdel 12436  ax-bdsb 12437  ax-bdsep 12499  ax-infvn 12560
This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-rab 2379  df-v 2635  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-nul 3303  df-sn 3472  df-pr 3473  df-uni 3676  df-int 3711  df-suc 4222  df-iom 4434  df-bdc 12456  df-bj-ind 12546
This theorem is referenced by: (None)
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