Theorem bdss 16890

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 16872	. . 3 |- BOUNDED y e. A
3	2	ax-bdal 16844	. 2 |- BOUNDED A. y e. x y e. A
4		dfss3 3236	. 2 |- ( x C_ A <-> A. y e. x y e. A )
5	3, 4	bd0r 16851	1 |- BOUNDED x C_ A

This proof depends on syntax axioms:   e. wcel 2209   A.wral 2528   C_ wss 3220  BOUNDED wbd 16838  BOUNDED wbdc 16866

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-11 1559  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220  ax-bd0 16839  ax-bdal 16844

This proof depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-ral 2533  df-in 3226  df-ss 3233  df-bdc 16867

This theorem is used by:  bdeq0  16893  bdcpw  16895  bdvsn  16900  bdop  16901  bdeqsuc  16907  bj-nntrans  16977  bj-omtrans  16982