Theorem bdvsn 14711

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 14707	. . . 4 ⊢ BOUNDED {𝑦}
2	1	bdss 14701	. . 3 ⊢ BOUNDED 𝑥 ⊆ {𝑦}
3		bdcv 14685	. . . 4 ⊢ BOUNDED 𝑥
4	3	bdsnss 14710	. . 3 ⊢ BOUNDED {𝑦} ⊆ 𝑥
5	2, 4	ax-bdan 14652	. 2 ⊢ BOUNDED (𝑥 ⊆ {𝑦} ∧ {𝑦} ⊆ 𝑥)
6		eqss 3172	. 2 ⊢ (𝑥 = {𝑦} ↔ (𝑥 ⊆ {𝑦} ∧ {𝑦} ⊆ 𝑥))
7	5, 6	bd0r 14662	1 ⊢ BOUNDED 𝑥 = {𝑦}

Syntax hints:   ∧ wa 104   = wceq 1353   ⊆ wss 3131  {csn 3594  BOUNDED wbd 14649

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159  ax-bd0 14650  ax-bdan 14652  ax-bdal 14655  ax-bdeq 14657  ax-bdel 14658  ax-bdsb 14659

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-v 2741  df-in 3137  df-ss 3144  df-sn 3600  df-bdc 14678

This theorem is referenced by:  bdop  14712