Theorem bdvsn 17066

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 17062	. . . 4 |- BOUNDED { y }
2	1	bdss 17056	. . 3 |- BOUNDED x C_ { y }
3		bdcv 17040	. . . 4 |- BOUNDED x
4	3	bdsnss 17065	. . 3 |- BOUNDED { y } C_ x
5	2, 4	ax-bdan 17007	. 2 |- BOUNDED ( x C_ { y } /\ { y } C_ x )
6		eqss 3263	. 2 |- ( x = { y } <-> ( x C_ { y } /\ { y } C_ x ) )
7	5, 6	bd0r 17017	1 |- BOUNDED x = { y }

This proof depends on syntax axioms:   /\ wa 104   = wceq 1402   C_ wss 3220   {csn 3709  BOUNDED wbd 17004

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 17005  ax-bdan 17007  ax-bdal 17010  ax-bdeq 17012  ax-bdel 17013  ax-bdsb 17014

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-ral 2533  df-v 2823  df-in 3226  df-ss 3233  df-sn 3715  df-bdc 17033

This theorem is used by:  bdop  17067