Theorem bdvsn 15371

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 15367	. . . 4 ⊢ BOUNDED {𝑦}
2	1	bdss 15361	. . 3 ⊢ BOUNDED 𝑥 ⊆ {𝑦}
3		bdcv 15345	. . . 4 ⊢ BOUNDED 𝑥
4	3	bdsnss 15370	. . 3 ⊢ BOUNDED {𝑦} ⊆ 𝑥
5	2, 4	ax-bdan 15312	. 2 ⊢ BOUNDED (𝑥 ⊆ {𝑦} ∧ {𝑦} ⊆ 𝑥)
6		eqss 3194	. 2 ⊢ (𝑥 = {𝑦} ↔ (𝑥 ⊆ {𝑦} ∧ {𝑦} ⊆ 𝑥))
7	5, 6	bd0r 15322	1 ⊢ BOUNDED 𝑥 = {𝑦}

Syntax hints:   ∧ wa 104   = wceq 1364   ⊆ wss 3153  {csn 3618  BOUNDED wbd 15309

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 2175  ax-bd0 15310  ax-bdan 15312  ax-bdal 15315  ax-bdeq 15317  ax-bdel 15318  ax-bdsb 15319

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-v 2762  df-in 3159  df-ss 3166  df-sn 3624  df-bdc 15338

This theorem is referenced by:  bdop  15372