Theorem bdcun 13049

Description: The union 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
bdcun	|- BOUNDED ( A u. B )

Proof of Theorem bdcun

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bdcdif.1	. . . . 5 |- BOUNDED A
2	1	bdeli 13033	. . . 4 |- BOUNDED x e. A
3		bdcdif.2	. . . . 5 |- BOUNDED B
4	3	bdeli 13033	. . . 4 |- BOUNDED x e. B
5	2, 4	ax-bdor 13003	. . 3 |- BOUNDED ( x e. A \/ x e. B )
6	5	bdcab 13036	. 2 |- BOUNDED { x | ( x e. A \/ x e. B ) }
7		df-un 3070	. 2 |- ( A u. B ) = { x | ( x e. A \/ x e. B ) }
8	6, 7	bdceqir 13031	1 |- BOUNDED ( A u. B )

Syntax hints:   \/ wo 697   e. wcel 1480   {cab 2123   u. cun 3064  BOUNDED wbdc 13027

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 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-17 1506  ax-ial 1514  ax-ext 2119  ax-bd0 13000  ax-bdor 13003  ax-bdsb 13009

This theorem depends on definitions:  df-bi 116  df-clab 2124  df-cleq 2130  df-clel 2133  df-un 3070  df-bdc 13028

This theorem is referenced by:  bdcpr  13058  bdctp  13059  bdcsuc  13067