Theorem bdcsuc 16201

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 16169	. . 3 ⊢ BOUNDED 𝑥
2		bdcsn 16191	. . 3 ⊢ BOUNDED {𝑥}
3	1, 2	bdcun 16183	. 2 ⊢ BOUNDED (𝑥 ∪ {𝑥})
4		df-suc 4461	. 2 ⊢ suc 𝑥 = (𝑥 ∪ {𝑥})
5	3, 4	bdceqir 16165	1 ⊢ BOUNDED suc 𝑥

Syntax hints:   ∪ cun 3195  {csn 3666  suc csuc 4455  BOUNDED wbdc 16161

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 1493  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-4 1556  ax-17 1572  ax-ial 1580  ax-ext 2211  ax-bd0 16134  ax-bdor 16137  ax-bdeq 16141  ax-bdel 16142  ax-bdsb 16143

This theorem depends on definitions:  df-bi 117  df-clab 2216  df-cleq 2222  df-clel 2225  df-un 3201  df-sn 3672  df-suc 4461  df-bdc 16162

This theorem is referenced by:  bdeqsuc  16202