Theorem bdss 15800

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 15782	. . 3 |- BOUNDED y e. A
3	2	ax-bdal 15754	. 2 |- BOUNDED A. y e. x y e. A
4		dfss3 3182	. 2 |- ( x C_ A <-> A. y e. x y e. A )
5	3, 4	bd0r 15761	1 |- BOUNDED x C_ A

Syntax hints:   e. wcel 2176   A.wral 2484   C_ wss 3166  BOUNDED wbd 15748  BOUNDED wbdc 15776

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-11 1529  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187  ax-bd0 15749  ax-bdal 15754

This theorem depends on definitions:  df-bi 117  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-ral 2489  df-in 3172  df-ss 3179  df-bdc 15777

This theorem is referenced by:  bdeq0  15803  bdcpw  15805  bdvsn  15810  bdop  15811  bdeqsuc  15817  bj-nntrans  15887  bj-omtrans  15892