Theorem bdss 15356

Description: The inclusion of a setvar in a bounded class is a bounded formula. Note: apparently, we cannot prove from the present axioms that equality of two bounded classes is a bounded formula. (Contributed by BJ, 3-Oct-2019.)

Hypothesis

Ref	Expression
bdss.1	⊢ BOUNDED 𝐴

Assertion

Ref	Expression
bdss	⊢ BOUNDED 𝑥 ⊆ 𝐴

Proof of Theorem bdss

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bdss.1	. . . 4 ⊢ BOUNDED 𝐴
2	1	bdeli 15338	. . 3 ⊢ BOUNDED 𝑦 ∈ 𝐴
3	2	ax-bdal 15310	. 2 ⊢ BOUNDED ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐴
4		dfss3 3169	. 2 ⊢ (𝑥 ⊆ 𝐴 ↔ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐴)
5	3, 4	bd0r 15317	1 ⊢ BOUNDED 𝑥 ⊆ 𝐴

Syntax hints:   ∈ wcel 2164  ∀wral 2472   ⊆ wss 3153  BOUNDED wbd 15304  BOUNDED wbdc 15332

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-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175  ax-bd0 15305  ax-bdal 15310

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-ral 2477  df-in 3159  df-ss 3166  df-bdc 15333

This theorem is referenced by:  bdeq0  15359  bdcpw  15361  bdvsn  15366  bdop  15367  bdeqsuc  15373  bj-nntrans  15443  bj-omtrans  15448