Theorem bdcun 12739

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 12723	. . . 4 |- BOUNDED x e. A
3		bdcdif.2	. . . . 5 |- BOUNDED B
4	3	bdeli 12723	. . . 4 |- BOUNDED x e. B
5	2, 4	ax-bdor 12693	. . 3 |- BOUNDED ( x e. A \/ x e. B )
6	5	bdcab 12726	. 2 |- BOUNDED { x | ( x e. A \/ x e. B ) }
7		df-un 3039	. 2 |- ( A u. B ) = { x | ( x e. A \/ x e. B ) }
8	6, 7	bdceqir 12721	1 |- BOUNDED ( A u. B )

Syntax hints:   \/ wo 680   e. wcel 1461   {cab 2099   u. cun 3033  BOUNDED wbdc 12717

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1404  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-4 1468  ax-17 1487  ax-ial 1495  ax-ext 2095  ax-bd0 12690  ax-bdor 12693  ax-bdsb 12699

This theorem depends on definitions:  df-bi 116  df-clab 2100  df-cleq 2106  df-clel 2109  df-un 3039  df-bdc 12718

This theorem is referenced by:  bdcpr  12748  bdctp  12749  bdcsuc  12757