Theorem bdvsn 13243

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 13239	. . . 4 ⊢ BOUNDED {𝑦}
2	1	bdss 13233	. . 3 ⊢ BOUNDED 𝑥 ⊆ {𝑦}
3		bdcv 13217	. . . 4 ⊢ BOUNDED 𝑥
4	3	bdsnss 13242	. . 3 ⊢ BOUNDED {𝑦} ⊆ 𝑥
5	2, 4	ax-bdan 13184	. 2 ⊢ BOUNDED (𝑥 ⊆ {𝑦} ∧ {𝑦} ⊆ 𝑥)
6		eqss 3117	. 2 ⊢ (𝑥 = {𝑦} ↔ (𝑥 ⊆ {𝑦} ∧ {𝑦} ⊆ 𝑥))
7	5, 6	bd0r 13194	1 ⊢ BOUNDED 𝑥 = {𝑦}

Syntax hints:   ∧ wa 103   = wceq 1332   ⊆ wss 3076  {csn 3532  BOUNDED wbd 13181

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-bd0 13182  ax-bdan 13184  ax-bdal 13187  ax-bdeq 13189  ax-bdel 13190  ax-bdsb 13191

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-v 2691  df-in 3082  df-ss 3089  df-sn 3538  df-bdc 13210

This theorem is referenced by:  bdop  13244