Theorem bdvsn 16167

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 16163	. . . 4 ⊢ BOUNDED {𝑦}
2	1	bdss 16157	. . 3 ⊢ BOUNDED 𝑥 ⊆ {𝑦}
3		bdcv 16141	. . . 4 ⊢ BOUNDED 𝑥
4	3	bdsnss 16166	. . 3 ⊢ BOUNDED {𝑦} ⊆ 𝑥
5	2, 4	ax-bdan 16108	. 2 ⊢ BOUNDED (𝑥 ⊆ {𝑦} ∧ {𝑦} ⊆ 𝑥)
6		eqss 3239	. 2 ⊢ (𝑥 = {𝑦} ↔ (𝑥 ⊆ {𝑦} ∧ {𝑦} ⊆ 𝑥))
7	5, 6	bd0r 16118	1 ⊢ BOUNDED 𝑥 = {𝑦}

Syntax hints:   ∧ wa 104   = wceq 1395   ⊆ wss 3197  {csn 3666  BOUNDED wbd 16105

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211  ax-bd0 16106  ax-bdan 16108  ax-bdal 16111  ax-bdeq 16113  ax-bdel 16114  ax-bdsb 16115

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-v 2801  df-in 3203  df-ss 3210  df-sn 3672  df-bdc 16134

This theorem is referenced by:  bdop  16168