Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >   Mathboxes  >  bj-bdind GIF version

Theorem bj-bdind 15140
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 15078 . . 3 BOUNDED ∅ ∈ 𝑥
2 bj-bdsucel 15092 . . . 4 BOUNDED suc 𝑦𝑥
32ax-bdal 15028 . . 3 BOUNDED𝑦𝑥 suc 𝑦𝑥
41, 3ax-bdan 15025 . 2 BOUNDED (∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥)
5 df-bj-ind 15137 . 2 (Ind 𝑥 ↔ (∅ ∈ 𝑥 ∧ ∀𝑦𝑥 suc 𝑦𝑥))
64, 5bd0r 15035 1 BOUNDED Ind 𝑥
Colors of variables: wff set class
Syntax hints:  wa 104  wcel 2160  wral 2468  c0 3437  suc csuc 4383  BOUNDED wbd 15022  Ind wind 15136
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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171  ax-bd0 15023  ax-bdim 15024  ax-bdan 15025  ax-bdor 15026  ax-bdn 15027  ax-bdal 15028  ax-bdex 15029  ax-bdeq 15030  ax-bdel 15031  ax-bdsb 15032
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-sn 3613  df-suc 4389  df-bdc 15051  df-bj-ind 15137
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator