Theorem bdvsn 10665

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 10661	. . . 4 |- BOUNDED { y }
2	1	bdss 10655	. . 3 |- BOUNDED x C_ { y }
3		bdcv 10639	. . . 4 |- BOUNDED x
4	3	bdsnss 10664	. . 3 |- BOUNDED { y } C_ x
5	2, 4	ax-bdan 10606	. 2 |- BOUNDED ( x C_ { y } /\ { y } C_ x )
6		eqss 3014	. 2 |- ( x = { y } <-> ( x C_ { y } /\ { y } C_ x ) )
7	5, 6	bd0r 10616	1 |- BOUNDED x = { y }

Syntax hints:   /\ wa 102   = wceq 1284   C_ wss 2973   {csn 3398  BOUNDED wbd 10603

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-bd0 10604  ax-bdan 10606  ax-bdal 10609  ax-bdeq 10611  ax-bdel 10612  ax-bdsb 10613

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-v 2603  df-in 2979  df-ss 2986  df-sn 3404  df-bdc 10632

This theorem is referenced by:  bdop  10666