Theorem bdss 17095

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 17077	. . 3 |- BOUNDED y e. A
3	2	ax-bdal 17049	. 2 |- BOUNDED A. y e. x y e. A
4		dfss3 3236	. 2 |- ( x C_ A <-> A. y e. x y e. A )
5	3, 4	bd0r 17056	1 |- BOUNDED x C_ A

This proof depends on syntax axioms:   e. wcel 2209   A.wral 2528   C_ wss 3220  BOUNDED wbd 17043  BOUNDED wbdc 17071

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-11 1559  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220  ax-bd0 17044  ax-bdal 17049

This proof depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-ral 2533  df-in 3226  df-ss 3233  df-bdc 17072

This theorem is used by:  bdeq0  17098  bdcpw  17100  bdvsn  17105  bdop  17106  bdeqsuc  17112  bj-nntrans  17182  bj-omtrans  17187