Theorem bdss 16580

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 16562	. . 3 |- BOUNDED y e. A
3	2	ax-bdal 16534	. 2 |- BOUNDED A. y e. x y e. A
4		dfss3 3217	. 2 |- ( x C_ A <-> A. y e. x y e. A )
5	3, 4	bd0r 16541	1 |- BOUNDED x C_ A

Syntax hints:   e. wcel 2202   A.wral 2511   C_ wss 3201  BOUNDED wbd 16528  BOUNDED wbdc 16556

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 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-11 1555  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213  ax-bd0 16529  ax-bdal 16534

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-ral 2516  df-in 3207  df-ss 3214  df-bdc 16557

This theorem is referenced by:  bdeq0  16583  bdcpw  16585  bdvsn  16590  bdop  16591  bdeqsuc  16597  bj-nntrans  16667  bj-omtrans  16672