Theorem bj-bdsucel 17006

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 17005	. 2 ⊢ BOUNDED 𝑧 = suc 𝑥
2	1	bj-bdcel 16961	1 ⊢ BOUNDED suc 𝑥 ∈ 𝑦

This proof depends on syntax axioms:   ∈ wcel 2209  suc csuc 4510  BOUNDED wbd 16936

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220  ax-bd0 16937  ax-bdan 16939  ax-bdor 16940  ax-bdal 16942  ax-bdex 16943  ax-bdeq 16944  ax-bdel 16945  ax-bdsb 16946

This proof depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-sn 3715  df-suc 4516  df-bdc 16965

This theorem is used by:  bj-bdind  17054