Theorem bdss 10655

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 10637	. . 3 ⊢ BOUNDED 𝑦 ∈ 𝐴
3	2	ax-bdal 10609	. 2 ⊢ BOUNDED ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐴
4		dfss3 2989	. 2 ⊢ (𝑥 ⊆ 𝐴 ↔ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐴)
5	3, 4	bd0r 10616	1 ⊢ BOUNDED 𝑥 ⊆ 𝐴

Syntax hints:   ∈ wcel 1433  ∀wral 2348   ⊆ wss 2973  BOUNDED wbd 10603  BOUNDED wbdc 10631

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-11 1437  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-bd0 10604  ax-bdal 10609

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-ral 2353  df-in 2979  df-ss 2986  df-bdc 10632

This theorem is referenced by:  bdeq0  10658  bdcpw  10660  bdvsn  10665  bdop  10666  bdeqsuc  10672  bj-nntrans  10746  bj-omtrans  10751