Theorem bdcsuc 16579

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 16547	. . 3 ⊢ BOUNDED 𝑥
2		bdcsn 16569	. . 3 ⊢ BOUNDED {𝑥}
3	1, 2	bdcun 16561	. 2 ⊢ BOUNDED (𝑥 ∪ {𝑥})
4		df-suc 4474	. 2 ⊢ suc 𝑥 = (𝑥 ∪ {𝑥})
5	3, 4	bdceqir 16543	1 ⊢ BOUNDED suc 𝑥

Syntax hints:   ∪ cun 3199  {csn 3673  suc csuc 4468  BOUNDED wbdc 16539

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 2213  ax-bd0 16512  ax-bdor 16515  ax-bdeq 16519  ax-bdel 16520  ax-bdsb 16521

This theorem depends on definitions:  df-bi 117  df-clab 2218  df-cleq 2224  df-clel 2227  df-un 3205  df-sn 3679  df-suc 4474  df-bdc 16540

This theorem is referenced by:  bdeqsuc  16580