Theorem bj-bdsucel 14905

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 14904	. 2 ⊢ BOUNDED 𝑧 = suc 𝑥
2	1	bj-bdcel 14860	1 ⊢ BOUNDED suc 𝑥 ∈ 𝑦

Syntax hints:   ∈ wcel 2158  suc csuc 4377  BOUNDED wbd 14835

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169  ax-bd0 14836  ax-bdan 14838  ax-bdor 14839  ax-bdal 14841  ax-bdex 14842  ax-bdeq 14843  ax-bdel 14844  ax-bdsb 14845

This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-v 2751  df-un 3145  df-in 3147  df-ss 3154  df-sn 3610  df-suc 4383  df-bdc 14864

This theorem is referenced by:  bj-bdind  14953