Theorem bdcdif 10652

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 10637	. . . 4 |- BOUNDED x e. A
3		bdcdif.2	. . . . . 6 |- BOUNDED B
4	3	bdeli 10637	. . . . 5 |- BOUNDED x e. B
5	4	ax-bdn 10608	. . . 4 |- BOUNDED -. x e. B
6	2, 5	ax-bdan 10606	. . 3 |- BOUNDED ( x e. A /\ -. x e. B )
7	6	bdcab 10640	. 2 |- BOUNDED { x | ( x e. A /\ -. x e. B ) }
8		df-dif 2975	. 2 |- ( A \ B ) = { x | ( x e. A /\ -. x e. B ) }
9	7, 8	bdceqir 10635	1 |- BOUNDED ( A \ B )

Syntax hints:   -. wn 3   /\ wa 102   e. wcel 1433   {cab 2067   \ cdif 2970  BOUNDED wbdc 10631

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-4 1440  ax-17 1459  ax-ial 1467  ax-ext 2063  ax-bd0 10604  ax-bdan 10606  ax-bdn 10608  ax-bdsb 10613

This theorem depends on definitions:  df-bi 115  df-clab 2068  df-cleq 2074  df-clel 2077  df-dif 2975  df-bdc 10632

This theorem is referenced by:  bdcnulALT  10657