Theorem bdss 13233

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 13215	. . 3 |- BOUNDED y e. A
3	2	ax-bdal 13187	. 2 |- BOUNDED A. y e. x y e. A
4		dfss3 3092	. 2 |- ( x C_ A <-> A. y e. x y e. A )
5	3, 4	bd0r 13194	1 |- BOUNDED x C_ A

Syntax hints:   e. wcel 1481   A.wral 2417   C_ wss 3076  BOUNDED wbd 13181  BOUNDED wbdc 13209

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-11 1485  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-bd0 13182  ax-bdal 13187

This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-ral 2422  df-in 3082  df-ss 3089  df-bdc 13210

This theorem is referenced by:  bdeq0  13236  bdcpw  13238  bdvsn  13243  bdop  13244  bdeqsuc  13250  bj-nntrans  13320  bj-omtrans  13325