Theorem bdcdif 9981

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 9966	. . . 4 |- BOUNDED x e. A
3		bdcdif.2	. . . . . 6 |- BOUNDED B
4	3	bdeli 9966	. . . . 5 |- BOUNDED x e. B
5	4	ax-bdn 9937	. . . 4 |- BOUNDED -. x e. B
6	2, 5	ax-bdan 9935	. . 3 |- BOUNDED ( x e. A /\ -. x e. B )
7	6	bdcab 9969	. 2 |- BOUNDED { x | ( x e. A /\ -. x e. B ) }
8		df-dif 2920	. 2 |- ( A \ B ) = { x | ( x e. A /\ -. x e. B ) }
9	7, 8	bdceqir 9964	1 |- BOUNDED ( A \ B )

Syntax hints:   -. wn 3   /\ wa 97   e. wcel 1393   {cab 2026   \ cdif 2914  BOUNDED wbdc 9960

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-17 1419  ax-ial 1427  ax-ext 2022  ax-bd0 9933  ax-bdan 9935  ax-bdn 9937  ax-bdsb 9942

This theorem depends on definitions:  df-bi 110  df-clab 2027  df-cleq 2033  df-clel 2036  df-dif 2920  df-bdc 9961

This theorem is referenced by:  bdcnulALT  9986