Theorem bdsnss 16146

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 16119	. 2 ⊢ BOUNDED 𝑥 ∈ 𝐴
3		vex 2782	. . 3 ⊢ 𝑥 ∈ V
4	3	snss 3782	. 2 ⊢ (𝑥 ∈ 𝐴 ↔ {𝑥} ⊆ 𝐴)
5	2, 4	bd0 16097	1 ⊢ BOUNDED {𝑥} ⊆ 𝐴

Syntax hints:   ∈ wcel 2180   ⊆ wss 3177  {csn 3646  BOUNDED wbd 16085  BOUNDED wbdc 16113

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 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-ext 2191  ax-bd0 16086

This theorem depends on definitions:  df-bi 117  df-tru 1378  df-nf 1487  df-sb 1789  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-v 2781  df-in 3183  df-ss 3190  df-sn 3652  df-bdc 16114

This theorem is referenced by:  bdvsn  16147  bdeqsuc  16154