Theorem bdvsn 7101

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 7097	. . . 4 ⊢ BOUNDED {y}
2	1	bdss 7091	. . 3 ⊢ BOUNDED x ⊆ {y}
3		bdcv 7075	. . . 4 ⊢ BOUNDED x
4	3	bdsnss 7100	. . 3 ⊢ BOUNDED {y} ⊆ x
5	2, 4	ax-bdan 7042	. 2 ⊢ BOUNDED (x ⊆ {y} ∧ {y} ⊆ x)
6		eqss 2937	. 2 ⊢ (x = {y} ↔ (x ⊆ {y} ∧ {y} ⊆ x))
7	5, 6	bd0r 7052	1 ⊢ BOUNDED x = {y}

Syntax hints:   ∧ wa 97   = wceq 1228   ⊆ wss 2894  {csn 3350  BOUNDED wbd 7039

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 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-11 1378  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-bd0 7040  ax-bdan 7042  ax-bdal 7045  ax-bdeq 7047  ax-bdel 7048  ax-bdsb 7049

This theorem depends on definitions:  df-bi 110  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-ral 2289  df-v 2537  df-in 2901  df-ss 2908  df-sn 3356  df-bdc 7068

This theorem is referenced by:  bdop  7102