Theorem bdcin 10654

Description: The intersection 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
bdcin	|- BOUNDED ( A i^i B )

Proof of Theorem bdcin

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	. . . . 5 |- BOUNDED B
4	3	bdeli 10637	. . . 4 |- BOUNDED x e. B
5	2, 4	ax-bdan 10606	. . 3 |- BOUNDED ( x e. A /\ x e. B )
6	5	bdcab 10640	. 2 |- BOUNDED { x | ( x e. A /\ x e. B ) }
7		df-in 2979	. 2 |- ( A i^i B ) = { x | ( x e. A /\ x e. B ) }
8	6, 7	bdceqir 10635	1 |- BOUNDED ( A i^i B )

Syntax hints:   /\ wa 102   e. wcel 1433   {cab 2067   i^i cin 2972  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-bdsb 10613

This theorem depends on definitions:  df-bi 115  df-clab 2068  df-cleq 2074  df-clel 2077  df-in 2979  df-bdc 10632

This theorem is referenced by: (None)