Theorem bdcdif 13753

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 13738	. . . 4 |- BOUNDED x e. A
3		bdcdif.2	. . . . . 6 |- BOUNDED B
4	3	bdeli 13738	. . . . 5 |- BOUNDED x e. B
5	4	ax-bdn 13709	. . . 4 |- BOUNDED -. x e. B
6	2, 5	ax-bdan 13707	. . 3 |- BOUNDED ( x e. A /\ -. x e. B )
7	6	bdcab 13741	. 2 |- BOUNDED { x | ( x e. A /\ -. x e. B ) }
8		df-dif 3118	. 2 |- ( A \ B ) = { x | ( x e. A /\ -. x e. B ) }
9	7, 8	bdceqir 13736	1 |- BOUNDED ( A \ B )

Syntax hints:   -. wn 3   /\ wa 103   e. wcel 2136   {cab 2151   \ cdif 3113  BOUNDED wbdc 13732

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-4 1498  ax-17 1514  ax-ial 1522  ax-ext 2147  ax-bd0 13705  ax-bdan 13707  ax-bdn 13709  ax-bdsb 13714

This theorem depends on definitions:  df-bi 116  df-clab 2152  df-cleq 2158  df-clel 2161  df-dif 3118  df-bdc 13733

This theorem is referenced by:  bdcnulALT  13758