Theorem bdcdif 15730

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 15715	. . . 4 |- BOUNDED x e. A
3		bdcdif.2	. . . . . 6 |- BOUNDED B
4	3	bdeli 15715	. . . . 5 |- BOUNDED x e. B
5	4	ax-bdn 15686	. . . 4 |- BOUNDED -. x e. B
6	2, 5	ax-bdan 15684	. . 3 |- BOUNDED ( x e. A /\ -. x e. B )
7	6	bdcab 15718	. 2 |- BOUNDED { x | ( x e. A /\ -. x e. B ) }
8		df-dif 3167	. 2 |- ( A \ B ) = { x | ( x e. A /\ -. x e. B ) }
9	7, 8	bdceqir 15713	1 |- BOUNDED ( A \ B )

Syntax hints:   -. wn 3   /\ wa 104   e. wcel 2175   {cab 2190   \ cdif 3162  BOUNDED wbdc 15709

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 1469  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-4 1532  ax-17 1548  ax-ial 1556  ax-ext 2186  ax-bd0 15682  ax-bdan 15684  ax-bdn 15686  ax-bdsb 15691

This theorem depends on definitions:  df-bi 117  df-clab 2191  df-cleq 2197  df-clel 2200  df-dif 3167  df-bdc 15710

This theorem is referenced by:  bdcnulALT  15735