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Theorem bdfind 14925
Description: Bounded induction (principle of induction when 𝐴 is assumed to be bounded), proved from basic constructive axioms. See find 4610 for a nonconstructive proof of the general case. See findset 14924 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 14921 . . . 4 ω ∈ V
31, 2bdssex 14881 . . 3 (𝐴 ⊆ ω → 𝐴 ∈ V)
433ad2ant1 1019 . 2 ((𝐴 ⊆ ω ∧ ∅ ∈ 𝐴 ∧ ∀𝑥𝐴 suc 𝑥𝐴) → 𝐴 ∈ V)
5 findset 14924 . 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 979   = wceq 1363  wcel 2158  wral 2465  Vcvv 2749  wss 3141  c0 3434  suc csuc 4377  ωcom 4601  BOUNDED wbdc 14819
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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-nul 4141  ax-pr 4221  ax-un 4445  ax-bd0 14792  ax-bdan 14794  ax-bdor 14795  ax-bdex 14798  ax-bdeq 14799  ax-bdel 14800  ax-bdsb 14801  ax-bdsep 14863  ax-infvn 14920
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-rab 2474  df-v 2751  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-sn 3610  df-pr 3611  df-uni 3822  df-int 3857  df-suc 4383  df-iom 4602  df-bdc 14820  df-bj-ind 14906
This theorem is referenced by: (None)
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