Theorem bdcin 16633

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 16616	. . . 4 |- BOUNDED x e. A
3		bdcdif.2	. . . . 5 |- BOUNDED B
4	3	bdeli 16616	. . . 4 |- BOUNDED x e. B
5	2, 4	ax-bdan 16585	. . 3 |- BOUNDED ( x e. A /\ x e. B )
6	5	bdcab 16619	. 2 |- BOUNDED { x | ( x e. A /\ x e. B ) }
7		df-in 3217	. 2 |- ( A i^i B ) = { x | ( x e. A /\ x e. B ) }
8	6, 7	bdceqir 16614	1 |- BOUNDED ( A i^i B )

Syntax hints:   /\ wa 104   e. wcel 2203   {cab 2218   i^i cin 3210  BOUNDED wbdc 16610

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1496  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-4 1559  ax-17 1575  ax-ial 1583  ax-ext 2214  ax-bd0 16583  ax-bdan 16585  ax-bdsb 16592

This theorem depends on definitions:  df-bi 117  df-clab 2219  df-cleq 2225  df-clel 2228  df-in 3217  df-bdc 16611

This theorem is referenced by: (None)