Theorem bdcsn 13057

Description: The singleton of a setvar is bounded. (Contributed by BJ, 16-Oct-2019.)

Assertion

Ref	Expression
bdcsn	⊢ BOUNDED {𝑥}

Proof of Theorem bdcsn

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-bdeq 13007	. . 3 ⊢ BOUNDED 𝑦 = 𝑥
2	1	bdcab 13036	. 2 ⊢ BOUNDED {𝑦 ∣ 𝑦 = 𝑥}
3		df-sn 3528	. 2 ⊢ {𝑥} = {𝑦 ∣ 𝑦 = 𝑥}
4	2, 3	bdceqir 13031	1 ⊢ BOUNDED {𝑥}

Syntax hints:  {cab 2123  {csn 3522  BOUNDED wbdc 13027

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-17 1506  ax-ial 1514  ax-ext 2119  ax-bd0 13000  ax-bdeq 13007  ax-bdsb 13009

This theorem depends on definitions:  df-bi 116  df-clab 2124  df-cleq 2130  df-clel 2133  df-sn 3528  df-bdc 13028

This theorem is referenced by:  bdcpr  13058  bdctp  13059  bdvsn  13061  bdcsuc  13067