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Theorem bj-bdind 13212
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 13150 . . 3 BOUNDED ∅ ∈ 𝑥
2 bj-bdsucel 13164 . . . 4 BOUNDED suc 𝑦𝑥
32ax-bdal 13100 . . 3 BOUNDED𝑦𝑥 suc 𝑦𝑥
41, 3ax-bdan 13097 . 2 BOUNDED (∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥)
5 df-bj-ind 13209 . 2 (Ind 𝑥 ↔ (∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥))
64, 5bd0r 13107 1 BOUNDED Ind 𝑥
Colors of variables: wff set class
Syntax hints:  wa 103  wcel 1480  wral 2416  c0 3363  suc csuc 4287  BOUNDED wbd 13094  Ind wind 13208
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-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-bd0 13095  ax-bdim 13096  ax-bdan 13097  ax-bdor 13098  ax-bdn 13099  ax-bdal 13100  ax-bdex 13101  ax-bdeq 13102  ax-bdel 13103  ax-bdsb 13104
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-fal 1337  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-v 2688  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-sn 3533  df-suc 4293  df-bdc 13123  df-bj-ind 13209
This theorem is referenced by: (None)
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