Theorem bdss 13746

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 13728	. . 3 |- BOUNDED y e. A
3	2	ax-bdal 13700	. 2 |- BOUNDED A. y e. x y e. A
4		dfss3 3132	. 2 |- ( x C_ A <-> A. y e. x y e. A )
5	3, 4	bd0r 13707	1 |- BOUNDED x C_ A

Syntax hints:   e. wcel 2136   A.wral 2444   C_ wss 3116  BOUNDED wbd 13694  BOUNDED wbdc 13722

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-11 1494  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147  ax-bd0 13695  ax-bdal 13700

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-ral 2449  df-in 3122  df-ss 3129  df-bdc 13723

This theorem is referenced by:  bdeq0  13749  bdcpw  13751  bdvsn  13756  bdop  13757  bdeqsuc  13763  bj-nntrans  13833  bj-omtrans  13838