Theorem bdcsuc 16475

Description: The successor of a setvar is a bounded class. (Contributed by BJ, 16-Oct-2019.)

Assertion

Ref	Expression
bdcsuc	⊢ BOUNDED suc 𝑥

Proof of Theorem bdcsuc

Step	Hyp	Ref	Expression
1		bdcv 16443	. . 3 ⊢ BOUNDED 𝑥
2		bdcsn 16465	. . 3 ⊢ BOUNDED {𝑥}
3	1, 2	bdcun 16457	. 2 ⊢ BOUNDED (𝑥 ∪ {𝑥})
4		df-suc 4468	. 2 ⊢ suc 𝑥 = (𝑥 ∪ {𝑥})
5	3, 4	bdceqir 16439	1 ⊢ BOUNDED suc 𝑥

Syntax hints:   ∪ cun 3198  {csn 3669  suc csuc 4462  BOUNDED wbdc 16435

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-5 1495  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-4 1558  ax-17 1574  ax-ial 1582  ax-ext 2213  ax-bd0 16408  ax-bdor 16411  ax-bdeq 16415  ax-bdel 16416  ax-bdsb 16417

This theorem depends on definitions:  df-bi 117  df-clab 2218  df-cleq 2224  df-clel 2227  df-un 3204  df-sn 3675  df-suc 4468  df-bdc 16436

This theorem is referenced by:  bdeqsuc  16476