Theorem bdcsn 16191

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 16141	. . 3 ⊢ BOUNDED 𝑦 = 𝑥
2	1	bdcab 16170	. 2 ⊢ BOUNDED {𝑦 ∣ 𝑦 = 𝑥}
3		df-sn 3672	. 2 ⊢ {𝑥} = {𝑦 ∣ 𝑦 = 𝑥}
4	2, 3	bdceqir 16165	1 ⊢ BOUNDED {𝑥}

Syntax hints:  {cab 2215  {csn 3666  BOUNDED wbdc 16161

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 1493  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-4 1556  ax-17 1572  ax-ial 1580  ax-ext 2211  ax-bd0 16134  ax-bdeq 16141  ax-bdsb 16143

This theorem depends on definitions:  df-bi 117  df-clab 2216  df-cleq 2222  df-clel 2225  df-sn 3672  df-bdc 16162

This theorem is referenced by:  bdcpr  16192  bdctp  16193  bdvsn  16195  bdcsuc  16201