Theorem bdss 15510

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 15492	. . 3 ⊢ BOUNDED 𝑦 ∈ 𝐴
3	2	ax-bdal 15464	. 2 ⊢ BOUNDED ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐴
4		dfss3 3173	. 2 ⊢ (𝑥 ⊆ 𝐴 ↔ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐴)
5	3, 4	bd0r 15471	1 ⊢ BOUNDED 𝑥 ⊆ 𝐴

Syntax hints:   ∈ wcel 2167  ∀wral 2475   ⊆ wss 3157  BOUNDED wbd 15458  BOUNDED wbdc 15486

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-11 1520  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178  ax-bd0 15459  ax-bdal 15464

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-ral 2480  df-in 3163  df-ss 3170  df-bdc 15487

This theorem is referenced by:  bdeq0  15513  bdcpw  15515  bdvsn  15520  bdop  15521  bdeqsuc  15527  bj-nntrans  15597  bj-omtrans  15602