Theorem bdss 13899

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 13881	. . 3 |- BOUNDED y e. A
3	2	ax-bdal 13853	. 2 |- BOUNDED A. y e. x y e. A
4		dfss3 3137	. 2 |- ( x C_ A <-> A. y e. x y e. A )
5	3, 4	bd0r 13860	1 |- BOUNDED x C_ A

Syntax hints:   e. wcel 2141   A.wral 2448   C_ wss 3121  BOUNDED wbd 13847  BOUNDED wbdc 13875

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 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-11 1499  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152  ax-bd0 13848  ax-bdal 13853

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-ral 2453  df-in 3127  df-ss 3134  df-bdc 13876

This theorem is referenced by:  bdeq0  13902  bdcpw  13904  bdvsn  13909  bdop  13910  bdeqsuc  13916  bj-nntrans  13986  bj-omtrans  13991