Theorem bdss 13984

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 13966	. . 3 ⊢ BOUNDED 𝑦 ∈ 𝐴
3	2	ax-bdal 13938	. 2 ⊢ BOUNDED ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐴
4		dfss3 3138	. 2 ⊢ (𝑥 ⊆ 𝐴 ↔ ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐴)
5	3, 4	bd0r 13945	1 ⊢ BOUNDED 𝑥 ⊆ 𝐴

Syntax hints:   ∈ wcel 2142  ∀wral 2449   ⊆ wss 3122  BOUNDED wbd 13932  BOUNDED wbdc 13960

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1441  ax-7 1442  ax-gen 1443  ax-ie1 1487  ax-ie2 1488  ax-8 1498  ax-11 1500  ax-4 1504  ax-17 1520  ax-i9 1524  ax-ial 1528  ax-i5r 1529  ax-ext 2153  ax-bd0 13933  ax-bdal 13938

This theorem depends on definitions:  df-bi 116  df-nf 1455  df-sb 1757  df-clab 2158  df-cleq 2164  df-clel 2167  df-ral 2454  df-in 3128  df-ss 3135  df-bdc 13961

This theorem is referenced by:  bdeq0  13987  bdcpw  13989  bdvsn  13994  bdop  13995  bdeqsuc  14001  bj-nntrans  14071  bj-omtrans  14076