Theorem bdss 9984

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 9966	. . 3 |- BOUNDED y e. A
3	2	ax-bdal 9938	. 2 |- BOUNDED A. y e. x y e. A
4		dfss3 2935	. 2 |- ( x C_ A <-> A. y e. x y e. A )
5	3, 4	bd0r 9945	1 |- BOUNDED x C_ A

Syntax hints:   e. wcel 1393   A.wral 2306   C_ wss 2917  BOUNDED wbd 9932  BOUNDED wbdc 9960

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-bd0 9933  ax-bdal 9938

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-ral 2311  df-in 2924  df-ss 2931  df-bdc 9961

This theorem is referenced by:  bdeq0  9987  bdcpw  9989  bdvsn  9994  bdop  9995  bdeqsuc  10001  bj-nntrans  10076  bj-omtrans  10081