Theorem bdcsuc 17111

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 17079	. . 3 ⊢ BOUNDED 𝑥
2		bdcsn 17101	. . 3 ⊢ BOUNDED {𝑥}
3	1, 2	bdcun 17093	. 2 ⊢ BOUNDED (𝑥 ∪ {𝑥})
4		df-suc 4518	. 2 ⊢ suc 𝑥 = (𝑥 ∪ {𝑥})
5	3, 4	bdceqir 17075	1 ⊢ BOUNDED suc 𝑥

This proof depends on syntax axioms:   ∪ cun 3218  {csn 3709  suc csuc 4512  BOUNDED wbdc 17071

This proof depends on 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 17044  ax-bdor 17047  ax-bdeq 17051  ax-bdel 17052  ax-bdsb 17053

This proof depends on definitions:  df-bi 117  df-clab 2225  df-cleq 2231  df-clel 2234  df-un 3224  df-sn 3715  df-suc 4518  df-bdc 17072

This theorem is used by:  bdeqsuc  17112