Theorem bdsnss 16528

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 16501	. 2 ⊢ BOUNDED 𝑥 ∈ 𝐴
3		vex 2804	. . 3 ⊢ 𝑥 ∈ V
4	3	snss 3809	. 2 ⊢ (𝑥 ∈ 𝐴 ↔ {𝑥} ⊆ 𝐴)
5	2, 4	bd0 16479	1 ⊢ BOUNDED {𝑥} ⊆ 𝐴

Syntax hints:   ∈ wcel 2201   ⊆ wss 3199  {csn 3670  BOUNDED wbd 16467  BOUNDED wbdc 16495

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2212  ax-bd0 16468

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1810  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-v 2803  df-in 3205  df-ss 3212  df-sn 3676  df-bdc 16496

This theorem is referenced by:  bdvsn  16529  bdeqsuc  16536