Theorem bj-bdsucel 15956

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 15955	. 2 ⊢ BOUNDED 𝑧 = suc 𝑥
2	1	bj-bdcel 15911	1 ⊢ BOUNDED suc 𝑥 ∈ 𝑦

Syntax hints:   ∈ wcel 2177  suc csuc 4420  BOUNDED wbd 15886

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188  ax-bd0 15887  ax-bdan 15889  ax-bdor 15890  ax-bdal 15892  ax-bdex 15893  ax-bdeq 15894  ax-bdel 15895  ax-bdsb 15896

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-v 2775  df-un 3174  df-in 3176  df-ss 3183  df-sn 3644  df-suc 4426  df-bdc 15915

This theorem is referenced by:  bj-bdind  16004