Theorem bdss 16227

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 16209	. . 3 |- BOUNDED y e. A
3	2	ax-bdal 16181	. 2 |- BOUNDED A. y e. x y e. A
4		dfss3 3213	. 2 |- ( x C_ A <-> A. y e. x y e. A )
5	3, 4	bd0r 16188	1 |- BOUNDED x C_ A

Syntax hints:   e. wcel 2200   A.wral 2508   C_ wss 3197  BOUNDED wbd 16175  BOUNDED wbdc 16203

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 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-11 1552  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211  ax-bd0 16176  ax-bdal 16181

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-ral 2513  df-in 3203  df-ss 3210  df-bdc 16204

This theorem is referenced by:  bdeq0  16230  bdcpw  16232  bdvsn  16237  bdop  16238  bdeqsuc  16244  bj-nntrans  16314  bj-omtrans  16319