Theorem bdsnss 13435

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 13408	. 2 ⊢ BOUNDED 𝑥 ∈ 𝐴
3		vex 2715	. . 3 ⊢ 𝑥 ∈ V
4	3	snss 3685	. 2 ⊢ (𝑥 ∈ 𝐴 ↔ {𝑥} ⊆ 𝐴)
5	2, 4	bd0 13386	1 ⊢ BOUNDED {𝑥} ⊆ 𝐴

Syntax hints:   ∈ wcel 2128   ⊆ wss 3102  {csn 3560  BOUNDED wbd 13374  BOUNDED wbdc 13402

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139  ax-bd0 13375

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-in 3108  df-ss 3115  df-sn 3566  df-bdc 13403

This theorem is referenced by:  bdvsn  13436  bdeqsuc  13443