Theorem bj-bdsucel 15037

Description: Boundedness of the formula "the successor of the setvar 𝑥 belongs to the setvar 𝑦". (Contributed by BJ, 30-Nov-2019.)

Assertion

Ref	Expression
bj-bdsucel	⊢ BOUNDED suc 𝑥 ∈ 𝑦

Proof of Theorem bj-bdsucel

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bdeqsuc 15036	. 2 ⊢ BOUNDED 𝑧 = suc 𝑥
2	1	bj-bdcel 14992	1 ⊢ BOUNDED suc 𝑥 ∈ 𝑦

Syntax hints:   ∈ wcel 2160  suc csuc 4380  BOUNDED wbd 14967

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-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 14968  ax-bdan 14970  ax-bdor 14971  ax-bdal 14973  ax-bdex 14974  ax-bdeq 14975  ax-bdel 14976  ax-bdsb 14977

This theorem depends on definitions:  df-bi 117  df-tru 1367  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-un 3148  df-in 3150  df-ss 3157  df-sn 3613  df-suc 4386  df-bdc 14996

This theorem is referenced by:  bj-bdind  15085