Theorem bdss 14701

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 14683	. . 3 |- BOUNDED y e. A
3	2	ax-bdal 14655	. 2 |- BOUNDED A. y e. x y e. A
4		dfss3 3147	. 2 |- ( x C_ A <-> A. y e. x y e. A )
5	3, 4	bd0r 14662	1 |- BOUNDED x C_ A

Syntax hints:   e. wcel 2148   A.wral 2455   C_ wss 3131  BOUNDED wbd 14649  BOUNDED wbdc 14677

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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159  ax-bd0 14650  ax-bdal 14655

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-ral 2460  df-in 3137  df-ss 3144  df-bdc 14678

This theorem is referenced by:  bdeq0  14704  bdcpw  14706  bdvsn  14711  bdop  14712  bdeqsuc  14718  bj-nntrans  14788  bj-omtrans  14793