Theorem bdcsuc 15610

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 15578	. . 3 ⊢ BOUNDED 𝑥
2		bdcsn 15600	. . 3 ⊢ BOUNDED {𝑥}
3	1, 2	bdcun 15592	. 2 ⊢ BOUNDED (𝑥 ∪ {𝑥})
4		df-suc 4407	. 2 ⊢ suc 𝑥 = (𝑥 ∪ {𝑥})
5	3, 4	bdceqir 15574	1 ⊢ BOUNDED suc 𝑥

Syntax hints:   ∪ cun 3155  {csn 3623  suc csuc 4401  BOUNDED wbdc 15570

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 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-4 1524  ax-17 1540  ax-ial 1548  ax-ext 2178  ax-bd0 15543  ax-bdor 15546  ax-bdeq 15550  ax-bdel 15551  ax-bdsb 15552

This theorem depends on definitions:  df-bi 117  df-clab 2183  df-cleq 2189  df-clel 2192  df-un 3161  df-sn 3629  df-suc 4407  df-bdc 15571

This theorem is referenced by:  bdeqsuc  15611