Theorem bdcun 13579

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 13563	. . . 4 |- BOUNDED x e. A
3		bdcdif.2	. . . . 5 |- BOUNDED B
4	3	bdeli 13563	. . . 4 |- BOUNDED x e. B
5	2, 4	ax-bdor 13533	. . 3 |- BOUNDED ( x e. A \/ x e. B )
6	5	bdcab 13566	. 2 |- BOUNDED { x | ( x e. A \/ x e. B ) }
7		df-un 3115	. 2 |- ( A u. B ) = { x | ( x e. A \/ x e. B ) }
8	6, 7	bdceqir 13561	1 |- BOUNDED ( A u. B )

Syntax hints:   \/ wo 698   e. wcel 2135   {cab 2150   u. cun 3109  BOUNDED wbdc 13557

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 1434  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-4 1497  ax-17 1513  ax-ial 1521  ax-ext 2146  ax-bd0 13530  ax-bdor 13533  ax-bdsb 13539

This theorem depends on definitions:  df-bi 116  df-clab 2151  df-cleq 2157  df-clel 2160  df-un 3115  df-bdc 13558

This theorem is referenced by:  bdcpr  13588  bdctp  13589  bdcsuc  13597