Theorem bdcsuc 13393

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 13361	. . 3 ⊢ BOUNDED 𝑥
2		bdcsn 13383	. . 3 ⊢ BOUNDED {𝑥}
3	1, 2	bdcun 13375	. 2 ⊢ BOUNDED (𝑥 ∪ {𝑥})
4		df-suc 4326	. 2 ⊢ suc 𝑥 = (𝑥 ∪ {𝑥})
5	3, 4	bdceqir 13357	1 ⊢ BOUNDED suc 𝑥

Syntax hints:   ∪ cun 3096  {csn 3556  suc csuc 4320  BOUNDED wbdc 13353

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1424  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-4 1487  ax-17 1503  ax-ial 1511  ax-ext 2136  ax-bd0 13326  ax-bdor 13329  ax-bdeq 13333  ax-bdel 13334  ax-bdsb 13335

This theorem depends on definitions:  df-bi 116  df-clab 2141  df-cleq 2147  df-clel 2150  df-un 3102  df-sn 3562  df-suc 4326  df-bdc 13354

This theorem is referenced by:  bdeqsuc  13394