Theorem bdss 9319

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 C_ 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 9301	. . 3 |- BOUNDED y e. A
3	2	ax-bdal 9273	. 2 |- BOUNDED A.y e. x y e. A
4		dfss3 2929	. 2 |- (x C_ A <-> A.y e. x y e. A)
5	3, 4	bd0r 9280	1 |- BOUNDED x C_ A

Syntax hints:   e. wcel 1390  A.wral 2300   C_ wss 2911  BOUNDED wbd 9267  BOUNDED wbdc 9295

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-bd0 9268  ax-bdal 9273

This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-ral 2305  df-in 2918  df-ss 2925  df-bdc 9296

This theorem is referenced by:  bdeq0  9322  bdcpw  9324  bdvsn  9329  bdop  9330  bdeqsuc  9336  bj-nntrans  9411  bj-omtrans  9416