Theorem bdss 16459

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 16441	. . 3 |- BOUNDED y e. A
3	2	ax-bdal 16413	. 2 |- BOUNDED A. y e. x y e. A
4		dfss3 3216	. 2 |- ( x C_ A <-> A. y e. x y e. A )
5	3, 4	bd0r 16420	1 |- BOUNDED x C_ A

Syntax hints:   e. wcel 2202   A.wral 2510   C_ wss 3200  BOUNDED wbd 16407  BOUNDED wbdc 16435

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 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-11 1554  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213  ax-bd0 16408  ax-bdal 16413

This theorem depends on definitions:  df-bi 117  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-ral 2515  df-in 3206  df-ss 3213  df-bdc 16436

This theorem is referenced by:  bdeq0  16462  bdcpw  16464  bdvsn  16469  bdop  16470  bdeqsuc  16476  bj-nntrans  16546  bj-omtrans  16551