Theorem bdss 15934

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 15916	. . 3 ⊢ BOUNDED 𝑦 ∈ 𝐴
3	2	ax-bdal 15888	. 2 ⊢ BOUNDED ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐴
4		dfss3 3186	. 2 ⊢ (𝑥 ⊆ 𝐴 ↔ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐴)
5	3, 4	bd0r 15895	1 ⊢ BOUNDED 𝑥 ⊆ 𝐴

Syntax hints:   ∈ wcel 2177  ∀wral 2485   ⊆ wss 3170  BOUNDED wbd 15882  BOUNDED wbdc 15910

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-11 1530  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188  ax-bd0 15883  ax-bdal 15888

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-ral 2490  df-in 3176  df-ss 3183  df-bdc 15911

This theorem is referenced by:  bdeq0  15937  bdcpw  15939  bdvsn  15944  bdop  15945  bdeqsuc  15951  bj-nntrans  16021  bj-omtrans  16026