Theorem bdsnss 16899

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 16872	. 2 |- BOUNDED x e. A
3		vex 2824	. . 3 |- x e. _V
4	3	snss 3850	. 2 |- ( x e. A <-> { x } C_ A )
5	2, 4	bd0 16850	1 |- BOUNDED { x } C_ A

This proof depends on syntax axioms:   e. wcel 2209   C_ wss 3220   {csn 3709  BOUNDED wbd 16838  BOUNDED wbdc 16866

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220  ax-bd0 16839

This proof depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-in 3226  df-ss 3233  df-sn 3715  df-bdc 16867

This theorem is used by:  bdvsn  16900  bdeqsuc  16907