Theorem bdcsn 9990

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 9940	. . 3 ⊢ BOUNDED 𝑦 = 𝑥
2	1	bdcab 9969	. 2 ⊢ BOUNDED {𝑦 ∣ 𝑦 = 𝑥}
3		df-sn 3381	. 2 ⊢ {𝑥} = {𝑦 ∣ 𝑦 = 𝑥}
4	2, 3	bdceqir 9964	1 ⊢ BOUNDED {𝑥}

Syntax hints:  {cab 2026  {csn 3375  BOUNDED wbdc 9960

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-17 1419  ax-ial 1427  ax-ext 2022  ax-bd0 9933  ax-bdeq 9940  ax-bdsb 9942

This theorem depends on definitions:  df-bi 110  df-clab 2027  df-cleq 2033  df-clel 2036  df-sn 3381  df-bdc 9961

This theorem is referenced by:  bdcpr  9991  bdctp  9992  bdvsn  9994  bdcsuc  10000