Theorem bdvsn 15604

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 15600	. . . 4 |- BOUNDED { y }
2	1	bdss 15594	. . 3 |- BOUNDED x C_ { y }
3		bdcv 15578	. . . 4 |- BOUNDED x
4	3	bdsnss 15603	. . 3 |- BOUNDED { y } C_ x
5	2, 4	ax-bdan 15545	. 2 |- BOUNDED ( x C_ { y } /\ { y } C_ x )
6		eqss 3199	. 2 |- ( x = { y } <-> ( x C_ { y } /\ { y } C_ x ) )
7	5, 6	bd0r 15555	1 |- BOUNDED x = { y }

Syntax hints:   /\ wa 104   = wceq 1364   C_ wss 3157   {csn 3623  BOUNDED wbd 15542

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178  ax-bd0 15543  ax-bdan 15545  ax-bdal 15548  ax-bdeq 15550  ax-bdel 15551  ax-bdsb 15552

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-v 2765  df-in 3163  df-ss 3170  df-sn 3629  df-bdc 15571

This theorem is referenced by:  bdop  15605