Theorem bdsnss 16630

Description: Inclusion of a singleton of a setvar in a bounded class is a bounded formula. (Contributed by BJ, 16-Oct-2019.)

Hypothesis

Ref	Expression
bdsnss.1	⊢ BOUNDED 𝐴

Assertion

Ref	Expression
bdsnss	⊢ BOUNDED {𝑥} ⊆ 𝐴

Distinct variable group:   𝑥,𝐴

Proof of Theorem bdsnss

Step	Hyp	Ref	Expression
1		bdsnss.1	. . 3 ⊢ BOUNDED 𝐴
2	1	bdeli 16603	. 2 ⊢ BOUNDED 𝑥 ∈ 𝐴
3		vex 2815	. . 3 ⊢ 𝑥 ∈ V
4	3	snss 3828	. 2 ⊢ (𝑥 ∈ 𝐴 ↔ {𝑥} ⊆ 𝐴)
5	2, 4	bd0 16581	1 ⊢ BOUNDED {𝑥} ⊆ 𝐴

Syntax hints:   ∈ wcel 2203   ⊆ wss 3210  {csn 3688  BOUNDED wbd 16569  BOUNDED wbdc 16597

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 2214  ax-bd0 16570

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2814  df-in 3216  df-ss 3223  df-sn 3694  df-bdc 16598

This theorem is referenced by:  bdvsn  16631  bdeqsuc  16638