Theorem bdcsn 15432

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 15382	. . 3 ⊢ BOUNDED 𝑦 = 𝑥
2	1	bdcab 15411	. 2 ⊢ BOUNDED {𝑦 ∣ 𝑦 = 𝑥}
3		df-sn 3625	. 2 ⊢ {𝑥} = {𝑦 ∣ 𝑦 = 𝑥}
4	2, 3	bdceqir 15406	1 ⊢ BOUNDED {𝑥}

Syntax hints:  {cab 2179  {csn 3619  BOUNDED wbdc 15402

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 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-4 1521  ax-17 1537  ax-ial 1545  ax-ext 2175  ax-bd0 15375  ax-bdeq 15382  ax-bdsb 15384

This theorem depends on definitions:  df-bi 117  df-clab 2180  df-cleq 2186  df-clel 2189  df-sn 3625  df-bdc 15403

This theorem is referenced by:  bdcpr  15433  bdctp  15434  bdvsn  15436  bdcsuc  15442