Theorem bdvsn 15810

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 15806	. . . 4 |- BOUNDED { y }
2	1	bdss 15800	. . 3 |- BOUNDED x C_ { y }
3		bdcv 15784	. . . 4 |- BOUNDED x
4	3	bdsnss 15809	. . 3 |- BOUNDED { y } C_ x
5	2, 4	ax-bdan 15751	. 2 |- BOUNDED ( x C_ { y } /\ { y } C_ x )
6		eqss 3208	. 2 |- ( x = { y } <-> ( x C_ { y } /\ { y } C_ x ) )
7	5, 6	bd0r 15761	1 |- BOUNDED x = { y }

Syntax hints:   /\ wa 104   = wceq 1373   C_ wss 3166   {csn 3633  BOUNDED wbd 15748

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187  ax-bd0 15749  ax-bdan 15751  ax-bdal 15754  ax-bdeq 15756  ax-bdel 15757  ax-bdsb 15758

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-v 2774  df-in 3172  df-ss 3179  df-sn 3639  df-bdc 15777

This theorem is referenced by:  bdop  15811