Theorem bdvsn 14166

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

Assertion

Ref	Expression
bdvsn	|- BOUNDED x = { y }

Distinct variable group:   x, y

Proof of Theorem bdvsn

Step	Hyp	Ref	Expression
1		bdcsn 14162	. . . 4 |- BOUNDED { y }
2	1	bdss 14156	. . 3 |- BOUNDED x C_ { y }
3		bdcv 14140	. . . 4 |- BOUNDED x
4	3	bdsnss 14165	. . 3 |- BOUNDED { y } C_ x
5	2, 4	ax-bdan 14107	. 2 |- BOUNDED ( x C_ { y } /\ { y } C_ x )
6		eqss 3168	. 2 |- ( x = { y } <-> ( x C_ { y } /\ { y } C_ x ) )
7	5, 6	bd0r 14117	1 |- BOUNDED x = { y }

Syntax hints:   /\ wa 104   = wceq 1353   C_ wss 3127   {csn 3589  BOUNDED wbd 14104

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 14105  ax-bdan 14107  ax-bdal 14110  ax-bdeq 14112  ax-bdel 14113  ax-bdsb 14114

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-ral 2458  df-v 2737  df-in 3133  df-ss 3140  df-sn 3595  df-bdc 14133

This theorem is referenced by:  bdop  14167