Theorem bdsnss 16166

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 16139	. 2 |- BOUNDED x e. A
3		vex 2802	. . 3 |- x e. _V
4	3	snss 3802	. 2 |- ( x e. A <-> { x } C_ A )
5	2, 4	bd0 16117	1 |- BOUNDED { x } C_ A

Syntax hints:   e. wcel 2200   C_ wss 3197   {csn 3666  BOUNDED wbd 16105  BOUNDED wbdc 16133

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211  ax-bd0 16106

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-in 3203  df-ss 3210  df-sn 3672  df-bdc 16134

This theorem is referenced by:  bdvsn  16167  bdeqsuc  16174