Theorem bdsnss 15365

Description: Inclusion of a singleton of a setvar in a bounded class is a bounded formula. (Contributed by BJ, 16-Oct-2019.)

Hypothesis

Ref	Expression
bdsnss.1	|- BOUNDED A

Assertion

Ref	Expression
bdsnss	|- BOUNDED { x } C_ A

Distinct variable group:   x, A

Proof of Theorem bdsnss

Step	Hyp	Ref	Expression
1		bdsnss.1	. . 3 |- BOUNDED A
2	1	bdeli 15338	. 2 |- BOUNDED x e. A
3		vex 2763	. . 3 |- x e. _V
4	3	snss 3753	. 2 |- ( x e. A <-> { x } C_ A )
5	2, 4	bd0 15316	1 |- BOUNDED { x } C_ A

Syntax hints:   e. wcel 2164   C_ wss 3153   {csn 3618  BOUNDED wbd 15304  BOUNDED wbdc 15332

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-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175  ax-bd0 15305

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-in 3159  df-ss 3166  df-sn 3624  df-bdc 15333

This theorem is referenced by:  bdvsn  15366  bdeqsuc  15373