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Theorem bj-bdind 13055
Description: Boundedness of the formula "the setvar 𝑥 is an inductive class". (Contributed by BJ, 30-Nov-2019.)
Assertion
Ref Expression
bj-bdind BOUNDED Ind 𝑥

Proof of Theorem bj-bdind
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 bj-bd0el 12993 . . 3 BOUNDED ∅ ∈ 𝑥
2 bj-bdsucel 13007 . . . 4 BOUNDED suc 𝑦𝑥
32ax-bdal 12943 . . 3 BOUNDED𝑦𝑥 suc 𝑦𝑥
41, 3ax-bdan 12940 . 2 BOUNDED (∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥)
5 df-bj-ind 13052 . 2 (Ind 𝑥 ↔ (∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥))
64, 5bd0r 12950 1 BOUNDED Ind 𝑥
Colors of variables: wff set class
Syntax hints:  wa 103  wcel 1465  wral 2393  c0 3333  suc csuc 4257  BOUNDED wbd 12937  Ind wind 13051
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-bd0 12938  ax-bdim 12939  ax-bdan 12940  ax-bdor 12941  ax-bdn 12942  ax-bdal 12943  ax-bdex 12944  ax-bdeq 12945  ax-bdel 12946  ax-bdsb 12947
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-sn 3503  df-suc 4263  df-bdc 12966  df-bj-ind 13052
This theorem is referenced by: (None)
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