Theorem bdss 13062

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 13044	. . 3 |- BOUNDED y e. A
3	2	ax-bdal 13016	. 2 |- BOUNDED A. y e. x y e. A
4		dfss3 3087	. 2 |- ( x C_ A <-> A. y e. x y e. A )
5	3, 4	bd0r 13023	1 |- BOUNDED x C_ A

Syntax hints:   e. wcel 1480   A.wral 2416   C_ wss 3071  BOUNDED wbd 13010  BOUNDED wbdc 13038

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 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-bd0 13011  ax-bdal 13016

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-ral 2421  df-in 3077  df-ss 3084  df-bdc 13039

This theorem is referenced by:  bdeq0  13065  bdcpw  13067  bdvsn  13072  bdop  13073  bdeqsuc  13079  bj-nntrans  13149  bj-omtrans  13154