Theorem bdvsn 15112

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 15108	. . . 4 |- BOUNDED { y }
2	1	bdss 15102	. . 3 |- BOUNDED x C_ { y }
3		bdcv 15086	. . . 4 |- BOUNDED x
4	3	bdsnss 15111	. . 3 |- BOUNDED { y } C_ x
5	2, 4	ax-bdan 15053	. 2 |- BOUNDED ( x C_ { y } /\ { y } C_ x )
6		eqss 3185	. 2 |- ( x = { y } <-> ( x C_ { y } /\ { y } C_ x ) )
7	5, 6	bd0r 15063	1 |- BOUNDED x = { y }

Syntax hints:   /\ wa 104   = wceq 1364   C_ wss 3144   {csn 3610  BOUNDED wbd 15050

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171  ax-bd0 15051  ax-bdan 15053  ax-bdal 15056  ax-bdeq 15058  ax-bdel 15059  ax-bdsb 15060

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-v 2754  df-in 3150  df-ss 3157  df-sn 3616  df-bdc 15079

This theorem is referenced by:  bdop  15113