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Theorem bdfind 13174
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 13173 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 13170 . . . 4 ω ∈ V
31, 2bdssex 13130 . . 3 (𝐴 ⊆ ω → 𝐴 ∈ V)
433ad2ant1 1002 . 2 ((𝐴 ⊆ ω ∧ ∅ ∈ 𝐴 ∧ ∀𝑥𝐴 suc 𝑥𝐴) → 𝐴 ∈ V)
5 findset 13173 . 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 962   = wceq 1331  wcel 1480  wral 2416  Vcvv 2686  wss 3071  c0 3363  suc csuc 4287  ωcom 4504  BOUNDED wbdc 13068
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 13041  ax-bdan 13043  ax-bdor 13044  ax-bdex 13047  ax-bdeq 13048  ax-bdel 13049  ax-bdsb 13050  ax-bdsep 13112  ax-infvn 13169
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 13069  df-bj-ind 13155
This theorem is referenced by: (None)
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