Theorem bdss 15999

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 15981	. . 3 |- BOUNDED y e. A
3	2	ax-bdal 15953	. 2 |- BOUNDED A. y e. x y e. A
4		dfss3 3190	. 2 |- ( x C_ A <-> A. y e. x y e. A )
5	3, 4	bd0r 15960	1 |- BOUNDED x C_ A

Syntax hints:   e. wcel 2178   A.wral 2486   C_ wss 3174  BOUNDED wbd 15947  BOUNDED wbdc 15975

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-11 1530  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189  ax-bd0 15948  ax-bdal 15953

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-ral 2491  df-in 3180  df-ss 3187  df-bdc 15976

This theorem is referenced by:  bdeq0  16002  bdcpw  16004  bdvsn  16009  bdop  16010  bdeqsuc  16016  bj-nntrans  16086  bj-omtrans  16091