Theorem bdcsuc 15372

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 15340	. . 3 ⊢ BOUNDED 𝑥
2		bdcsn 15362	. . 3 ⊢ BOUNDED {𝑥}
3	1, 2	bdcun 15354	. 2 ⊢ BOUNDED (𝑥 ∪ {𝑥})
4		df-suc 4402	. 2 ⊢ suc 𝑥 = (𝑥 ∪ {𝑥})
5	3, 4	bdceqir 15336	1 ⊢ BOUNDED suc 𝑥

Syntax hints:   ∪ cun 3151  {csn 3618  suc csuc 4396  BOUNDED wbdc 15332

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 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-4 1521  ax-17 1537  ax-ial 1545  ax-ext 2175  ax-bd0 15305  ax-bdor 15308  ax-bdeq 15312  ax-bdel 15313  ax-bdsb 15314

This theorem depends on definitions:  df-bi 117  df-clab 2180  df-cleq 2186  df-clel 2189  df-un 3157  df-sn 3624  df-suc 4402  df-bdc 15333

This theorem is referenced by:  bdeqsuc  15373