Theorem bdcsuc 16650

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 16618	. . 3 ⊢ BOUNDED 𝑥
2		bdcsn 16640	. . 3 ⊢ BOUNDED {𝑥}
3	1, 2	bdcun 16632	. 2 ⊢ BOUNDED (𝑥 ∪ {𝑥})
4		df-suc 4492	. 2 ⊢ suc 𝑥 = (𝑥 ∪ {𝑥})
5	3, 4	bdceqir 16614	1 ⊢ BOUNDED suc 𝑥

Syntax hints:   ∪ cun 3209  {csn 3689  suc csuc 4486  BOUNDED wbdc 16610

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 1496  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-4 1559  ax-17 1575  ax-ial 1583  ax-ext 2214  ax-bd0 16583  ax-bdor 16586  ax-bdeq 16590  ax-bdel 16591  ax-bdsb 16592

This theorem depends on definitions:  df-bi 117  df-clab 2219  df-cleq 2225  df-clel 2228  df-un 3215  df-sn 3695  df-suc 4492  df-bdc 16611

This theorem is referenced by:  bdeqsuc  16651