Theorem bdsnss 14185

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 14158	. 2 |- BOUNDED x e. A
3		vex 2738	. . 3 |- x e. _V
4	3	snss 3724	. 2 |- ( x e. A <-> { x } C_ A )
5	2, 4	bd0 14136	1 |- BOUNDED { x } C_ A

Syntax hints:   e. wcel 2146   C_ wss 3127   {csn 3589  BOUNDED wbd 14124  BOUNDED wbdc 14152

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 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157  ax-bd0 14125

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-v 2737  df-in 3133  df-ss 3140  df-sn 3595  df-bdc 14153

This theorem is referenced by:  bdvsn  14186  bdeqsuc  14193