Theorem bdcdif 16182

Description: The difference of two bounded classes is bounded. (Contributed by BJ, 3-Oct-2019.)

Hypotheses

Ref	Expression
bdcdif.1	|- BOUNDED A
bdcdif.2	|- BOUNDED B

Assertion

Ref	Expression
bdcdif	|- BOUNDED ( A \ B )

Proof of Theorem bdcdif

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bdcdif.1	. . . . 5 |- BOUNDED A
2	1	bdeli 16167	. . . 4 |- BOUNDED x e. A
3		bdcdif.2	. . . . . 6 |- BOUNDED B
4	3	bdeli 16167	. . . . 5 |- BOUNDED x e. B
5	4	ax-bdn 16138	. . . 4 |- BOUNDED -. x e. B
6	2, 5	ax-bdan 16136	. . 3 |- BOUNDED ( x e. A /\ -. x e. B )
7	6	bdcab 16170	. 2 |- BOUNDED { x | ( x e. A /\ -. x e. B ) }
8		df-dif 3199	. 2 |- ( A \ B ) = { x | ( x e. A /\ -. x e. B ) }
9	7, 8	bdceqir 16165	1 |- BOUNDED ( A \ B )

Syntax hints:   -. wn 3   /\ wa 104   e. wcel 2200   {cab 2215   \ cdif 3194  BOUNDED wbdc 16161

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 1493  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-4 1556  ax-17 1572  ax-ial 1580  ax-ext 2211  ax-bd0 16134  ax-bdan 16136  ax-bdn 16138  ax-bdsb 16143

This theorem depends on definitions:  df-bi 117  df-clab 2216  df-cleq 2222  df-clel 2225  df-dif 3199  df-bdc 16162

This theorem is referenced by:  bdcnulALT  16187