Theorem bdcsuc 16820

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 16788	. . 3 ⊢ BOUNDED 𝑥
2		bdcsn 16810	. . 3 ⊢ BOUNDED {𝑥}
3	1, 2	bdcun 16802	. 2 ⊢ BOUNDED (𝑥 ∪ {𝑥})
4		df-suc 4511	. 2 ⊢ suc 𝑥 = (𝑥 ∪ {𝑥})
5	3, 4	bdceqir 16784	1 ⊢ BOUNDED suc 𝑥

Syntax hints:   ∪ cun 3218  {csn 3705  suc csuc 4505  BOUNDED wbdc 16780

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 1500  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-4 1563  ax-17 1579  ax-ial 1587  ax-ext 2220  ax-bd0 16753  ax-bdor 16756  ax-bdeq 16760  ax-bdel 16761  ax-bdsb 16762

This theorem depends on definitions:  df-bi 117  df-clab 2225  df-cleq 2231  df-clel 2234  df-un 3224  df-sn 3711  df-suc 4511  df-bdc 16781

This theorem is referenced by:  bdeqsuc  16821