Theorem bdcsuc 15749

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 15717	. . 3 ⊢ BOUNDED 𝑥
2		bdcsn 15739	. . 3 ⊢ BOUNDED {𝑥}
3	1, 2	bdcun 15731	. 2 ⊢ BOUNDED (𝑥 ∪ {𝑥})
4		df-suc 4417	. 2 ⊢ suc 𝑥 = (𝑥 ∪ {𝑥})
5	3, 4	bdceqir 15713	1 ⊢ BOUNDED suc 𝑥

Syntax hints:   ∪ cun 3163  {csn 3632  suc csuc 4411  BOUNDED wbdc 15709

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 1469  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-4 1532  ax-17 1548  ax-ial 1556  ax-ext 2186  ax-bd0 15682  ax-bdor 15685  ax-bdeq 15689  ax-bdel 15690  ax-bdsb 15691

This theorem depends on definitions:  df-bi 117  df-clab 2191  df-cleq 2197  df-clel 2200  df-un 3169  df-sn 3638  df-suc 4417  df-bdc 15710

This theorem is referenced by:  bdeqsuc  15750