Theorem bj-bdsucel 15612

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 15611	. 2 ⊢ BOUNDED 𝑧 = suc 𝑥
2	1	bj-bdcel 15567	1 ⊢ BOUNDED suc 𝑥 ∈ 𝑦

Syntax hints:   ∈ wcel 2167  suc csuc 4401  BOUNDED wbd 15542

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178  ax-bd0 15543  ax-bdan 15545  ax-bdor 15546  ax-bdal 15548  ax-bdex 15549  ax-bdeq 15550  ax-bdel 15551  ax-bdsb 15552

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-sn 3629  df-suc 4407  df-bdc 15571

This theorem is referenced by:  bj-bdind  15660