Theorem bdss 15155

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 15137	. . 3 |- BOUNDED y e. A
3	2	ax-bdal 15109	. 2 |- BOUNDED A. y e. x y e. A
4		dfss3 3164	. 2 |- ( x C_ A <-> A. y e. x y e. A )
5	3, 4	bd0r 15116	1 |- BOUNDED x C_ A

Syntax hints:   e. wcel 2160   A.wral 2468   C_ wss 3148  BOUNDED wbd 15103  BOUNDED wbdc 15131

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171  ax-bd0 15104  ax-bdal 15109

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-ral 2473  df-in 3154  df-ss 3161  df-bdc 15132

This theorem is referenced by:  bdeq0  15158  bdcpw  15160  bdvsn  15165  bdop  15166  bdeqsuc  15172  bj-nntrans  15242  bj-omtrans  15247