Theorem bdvsn 15079

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 15075	. . . 4 ⊢ BOUNDED {𝑦}
2	1	bdss 15069	. . 3 ⊢ BOUNDED 𝑥 ⊆ {𝑦}
3		bdcv 15053	. . . 4 ⊢ BOUNDED 𝑥
4	3	bdsnss 15078	. . 3 ⊢ BOUNDED {𝑦} ⊆ 𝑥
5	2, 4	ax-bdan 15020	. 2 ⊢ BOUNDED (𝑥 ⊆ {𝑦} ∧ {𝑦} ⊆ 𝑥)
6		eqss 3185	. 2 ⊢ (𝑥 = {𝑦} ↔ (𝑥 ⊆ {𝑦} ∧ {𝑦} ⊆ 𝑥))
7	5, 6	bd0r 15030	1 ⊢ BOUNDED 𝑥 = {𝑦}

Syntax hints:   ∧ wa 104   = wceq 1364   ⊆ wss 3144  {csn 3607  BOUNDED wbd 15017

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171  ax-bd0 15018  ax-bdan 15020  ax-bdal 15023  ax-bdeq 15025  ax-bdel 15026  ax-bdsb 15027

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-v 2754  df-in 3150  df-ss 3157  df-sn 3613  df-bdc 15046

This theorem is referenced by:  bdop  15080