Theorem bdcin 13745

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 13728	. . . 4 |- BOUNDED x e. A
3		bdcdif.2	. . . . 5 |- BOUNDED B
4	3	bdeli 13728	. . . 4 |- BOUNDED x e. B
5	2, 4	ax-bdan 13697	. . 3 |- BOUNDED ( x e. A /\ x e. B )
6	5	bdcab 13731	. 2 |- BOUNDED { x | ( x e. A /\ x e. B ) }
7		df-in 3122	. 2 |- ( A i^i B ) = { x | ( x e. A /\ x e. B ) }
8	6, 7	bdceqir 13726	1 |- BOUNDED ( A i^i B )

Syntax hints:   /\ wa 103   e. wcel 2136   {cab 2151   i^i cin 3115  BOUNDED wbdc 13722

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-4 1498  ax-17 1514  ax-ial 1522  ax-ext 2147  ax-bd0 13695  ax-bdan 13697  ax-bdsb 13704

This theorem depends on definitions:  df-bi 116  df-clab 2152  df-cleq 2158  df-clel 2161  df-in 3122  df-bdc 13723

This theorem is referenced by: (None)