Theorem bdcin 15509

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 15492	. . . 4 |- BOUNDED x e. A
3		bdcdif.2	. . . . 5 |- BOUNDED B
4	3	bdeli 15492	. . . 4 |- BOUNDED x e. B
5	2, 4	ax-bdan 15461	. . 3 |- BOUNDED ( x e. A /\ x e. B )
6	5	bdcab 15495	. 2 |- BOUNDED { x | ( x e. A /\ x e. B ) }
7		df-in 3163	. 2 |- ( A i^i B ) = { x | ( x e. A /\ x e. B ) }
8	6, 7	bdceqir 15490	1 |- BOUNDED ( A i^i B )

Syntax hints:   /\ wa 104   e. wcel 2167   {cab 2182   i^i cin 3156  BOUNDED wbdc 15486

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 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-4 1524  ax-17 1540  ax-ial 1548  ax-ext 2178  ax-bd0 15459  ax-bdan 15461  ax-bdsb 15468

This theorem depends on definitions:  df-bi 117  df-clab 2183  df-cleq 2189  df-clel 2192  df-in 3163  df-bdc 15487

This theorem is referenced by: (None)