Theorem bdvsn 11720

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 11716	. . . 4 ⊢ BOUNDED {𝑦}
2	1	bdss 11710	. . 3 ⊢ BOUNDED 𝑥 ⊆ {𝑦}
3		bdcv 11694	. . . 4 ⊢ BOUNDED 𝑥
4	3	bdsnss 11719	. . 3 ⊢ BOUNDED {𝑦} ⊆ 𝑥
5	2, 4	ax-bdan 11661	. 2 ⊢ BOUNDED (𝑥 ⊆ {𝑦} ∧ {𝑦} ⊆ 𝑥)
6		eqss 3040	. 2 ⊢ (𝑥 = {𝑦} ↔ (𝑥 ⊆ {𝑦} ∧ {𝑦} ⊆ 𝑥))
7	5, 6	bd0r 11671	1 ⊢ BOUNDED 𝑥 = {𝑦}

Syntax hints:   ∧ wa 102   = wceq 1289   ⊆ wss 2999  {csn 3446  BOUNDED wbd 11658

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-bd0 11659  ax-bdan 11661  ax-bdal 11664  ax-bdeq 11666  ax-bdel 11667  ax-bdsb 11668

This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-v 2621  df-in 3005  df-ss 3012  df-sn 3452  df-bdc 11687

This theorem is referenced by:  bdop  11721