Theorem bdcun 14699

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 14683	. . . 4 |- BOUNDED x e. A
3		bdcdif.2	. . . . 5 |- BOUNDED B
4	3	bdeli 14683	. . . 4 |- BOUNDED x e. B
5	2, 4	ax-bdor 14653	. . 3 |- BOUNDED ( x e. A \/ x e. B )
6	5	bdcab 14686	. 2 |- BOUNDED { x | ( x e. A \/ x e. B ) }
7		df-un 3135	. 2 |- ( A u. B ) = { x | ( x e. A \/ x e. B ) }
8	6, 7	bdceqir 14681	1 |- BOUNDED ( A u. B )

Syntax hints:   \/ wo 708   e. wcel 2148   {cab 2163   u. cun 3129  BOUNDED wbdc 14677

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 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-17 1526  ax-ial 1534  ax-ext 2159  ax-bd0 14650  ax-bdor 14653  ax-bdsb 14659

This theorem depends on definitions:  df-bi 117  df-clab 2164  df-cleq 2170  df-clel 2173  df-un 3135  df-bdc 14678

This theorem is referenced by:  bdcpr  14708  bdctp  14709  bdcsuc  14717