Theorem bdvsn 9329

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 9325	. . . 4 ⊢ BOUNDED {y}
2	1	bdss 9319	. . 3 ⊢ BOUNDED x ⊆ {y}
3		bdcv 9303	. . . 4 ⊢ BOUNDED x
4	3	bdsnss 9328	. . 3 ⊢ BOUNDED {y} ⊆ x
5	2, 4	ax-bdan 9270	. 2 ⊢ BOUNDED (x ⊆ {y} ∧ {y} ⊆ x)
6		eqss 2954	. 2 ⊢ (x = {y} ↔ (x ⊆ {y} ∧ {y} ⊆ x))
7	5, 6	bd0r 9280	1 ⊢ BOUNDED x = {y}

Syntax hints:   ∧ wa 97   = wceq 1242   ⊆ wss 2911  {csn 3367  BOUNDED wbd 9267

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-bd0 9268  ax-bdan 9270  ax-bdal 9273  ax-bdeq 9275  ax-bdel 9276  ax-bdsb 9277

This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-ral 2305  df-v 2553  df-in 2918  df-ss 2925  df-sn 3373  df-bdc 9296

This theorem is referenced by:  bdop  9330