Theorem bdcdif 16887

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 16872	. . . 4 |- BOUNDED x e. A
3		bdcdif.2	. . . . . 6 |- BOUNDED B
4	3	bdeli 16872	. . . . 5 |- BOUNDED x e. B
5	4	ax-bdn 16843	. . . 4 |- BOUNDED -. x e. B
6	2, 5	ax-bdan 16841	. . 3 |- BOUNDED ( x e. A /\ -. x e. B )
7	6	bdcab 16875	. 2 |- BOUNDED { x | ( x e. A /\ -. x e. B ) }
8		df-dif 3222	. 2 |- ( A \ B ) = { x | ( x e. A /\ -. x e. B ) }
9	7, 8	bdceqir 16870	1 |- BOUNDED ( A \ B )

This proof depends on syntax axioms:   -. wn 3   /\ wa 104   e. wcel 2209   {cab 2224   \ cdif 3217  BOUNDED wbdc 16866

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-4 1563  ax-17 1579  ax-ial 1587  ax-ext 2220  ax-bd0 16839  ax-bdan 16841  ax-bdn 16843  ax-bdsb 16848

This proof depends on definitions:  df-bi 117  df-clab 2225  df-cleq 2231  df-clel 2234  df-dif 3222  df-bdc 16867

This theorem is used by:  bdcnulALT  16892