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Theorem bj-bdind 11182
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 11116 . . 3 BOUNDED ∅ ∈ 𝑥
2 bj-bdsucel 11130 . . . 4 BOUNDED suc 𝑦𝑥
32ax-bdal 11066 . . 3 BOUNDED𝑦𝑥 suc 𝑦𝑥
41, 3ax-bdan 11063 . 2 BOUNDED (∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥)
5 df-bj-ind 11179 . 2 (Ind 𝑥 ↔ (∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥))
64, 5bd0r 11073 1 BOUNDED Ind 𝑥
Colors of variables: wff set class
Syntax hints:  wa 102  wcel 1436  wral 2355  c0 3272  suc csuc 4159  BOUNDED wbd 11060  Ind wind 11178
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-bd0 11061  ax-bdim 11062  ax-bdan 11063  ax-bdor 11064  ax-bdn 11065  ax-bdal 11066  ax-bdex 11067  ax-bdeq 11068  ax-bdel 11069  ax-bdsb 11070
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-v 2616  df-dif 2988  df-un 2990  df-in 2992  df-ss 2999  df-nul 3273  df-sn 3431  df-suc 4165  df-bdc 11089  df-bj-ind 11179
This theorem is referenced by: (None)
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