Theorem bdvsn 14979

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 14975	. . . 4 |- BOUNDED { y }
2	1	bdss 14969	. . 3 |- BOUNDED x C_ { y }
3		bdcv 14953	. . . 4 |- BOUNDED x
4	3	bdsnss 14978	. . 3 |- BOUNDED { y } C_ x
5	2, 4	ax-bdan 14920	. 2 |- BOUNDED ( x C_ { y } /\ { y } C_ x )
6		eqss 3182	. 2 |- ( x = { y } <-> ( x C_ { y } /\ { y } C_ x ) )
7	5, 6	bd0r 14930	1 |- BOUNDED x = { y }

Syntax hints:   /\ wa 104   = wceq 1363   C_ wss 3141   {csn 3604  BOUNDED wbd 14917

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169  ax-bd0 14918  ax-bdan 14920  ax-bdal 14923  ax-bdeq 14925  ax-bdel 14926  ax-bdsb 14927

This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-v 2751  df-in 3147  df-ss 3154  df-sn 3610  df-bdc 14946

This theorem is referenced by:  bdop  14980