Theorem bdvsn 16009

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 16005	. . . 4 |- BOUNDED { y }
2	1	bdss 15999	. . 3 |- BOUNDED x C_ { y }
3		bdcv 15983	. . . 4 |- BOUNDED x
4	3	bdsnss 16008	. . 3 |- BOUNDED { y } C_ x
5	2, 4	ax-bdan 15950	. 2 |- BOUNDED ( x C_ { y } /\ { y } C_ x )
6		eqss 3216	. 2 |- ( x = { y } <-> ( x C_ { y } /\ { y } C_ x ) )
7	5, 6	bd0r 15960	1 |- BOUNDED x = { y }

Syntax hints:   /\ wa 104   = wceq 1373   C_ wss 3174   {csn 3643  BOUNDED wbd 15947

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189  ax-bd0 15948  ax-bdan 15950  ax-bdal 15953  ax-bdeq 15955  ax-bdel 15956  ax-bdsb 15957

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-v 2778  df-in 3180  df-ss 3187  df-sn 3649  df-bdc 15976

This theorem is referenced by:  bdop  16010