Theorem bdsnss 13071

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 13044	. 2 |- BOUNDED x e. A
3		vex 2689	. . 3 |- x e. _V
4	3	snss 3649	. 2 |- ( x e. A <-> { x } C_ A )
5	2, 4	bd0 13022	1 |- BOUNDED { x } C_ A

Syntax hints:   e. wcel 1480   C_ wss 3071   {csn 3527  BOUNDED wbd 13010  BOUNDED wbdc 13038

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-bd0 13011

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-in 3077  df-ss 3084  df-sn 3533  df-bdc 13039

This theorem is referenced by:  bdvsn  13072  bdeqsuc  13079