Theorem bdsnss 15063

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 15036	. 2 ⊢ BOUNDED 𝑥 ∈ 𝐴
3		vex 2755	. . 3 ⊢ 𝑥 ∈ V
4	3	snss 3742	. 2 ⊢ (𝑥 ∈ 𝐴 ↔ {𝑥} ⊆ 𝐴)
5	2, 4	bd0 15014	1 ⊢ BOUNDED {𝑥} ⊆ 𝐴

Syntax hints:   ∈ wcel 2160   ⊆ wss 3144  {csn 3607  BOUNDED wbd 15002  BOUNDED wbdc 15030

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171  ax-bd0 15003

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-in 3150  df-ss 3157  df-sn 3613  df-bdc 15031

This theorem is referenced by:  bdvsn  15064  bdeqsuc  15071