Theorem bdvsn 16770

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 16766	. . . 4 ⊢ BOUNDED {𝑦}
2	1	bdss 16760	. . 3 ⊢ BOUNDED 𝑥 ⊆ {𝑦}
3		bdcv 16744	. . . 4 ⊢ BOUNDED 𝑥
4	3	bdsnss 16769	. . 3 ⊢ BOUNDED {𝑦} ⊆ 𝑥
5	2, 4	ax-bdan 16711	. 2 ⊢ BOUNDED (𝑥 ⊆ {𝑦} ∧ {𝑦} ⊆ 𝑥)
6		eqss 3257	. 2 ⊢ (𝑥 = {𝑦} ↔ (𝑥 ⊆ {𝑦} ∧ {𝑦} ⊆ 𝑥))
7	5, 6	bd0r 16721	1 ⊢ BOUNDED 𝑥 = {𝑦}

Syntax hints:   ∧ wa 104   = wceq 1398   ⊆ wss 3214  {csn 3694  BOUNDED wbd 16708

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216  ax-bd0 16709  ax-bdan 16711  ax-bdal 16714  ax-bdeq 16716  ax-bdel 16717  ax-bdsb 16718

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-v 2817  df-in 3220  df-ss 3227  df-sn 3700  df-bdc 16737

This theorem is referenced by:  bdop  16771