Theorem bdcun 15835

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 15819	. . . 4 |- BOUNDED x e. A
3		bdcdif.2	. . . . 5 |- BOUNDED B
4	3	bdeli 15819	. . . 4 |- BOUNDED x e. B
5	2, 4	ax-bdor 15789	. . 3 |- BOUNDED ( x e. A \/ x e. B )
6	5	bdcab 15822	. 2 |- BOUNDED { x | ( x e. A \/ x e. B ) }
7		df-un 3170	. 2 |- ( A u. B ) = { x | ( x e. A \/ x e. B ) }
8	6, 7	bdceqir 15817	1 |- BOUNDED ( A u. B )

Syntax hints:   \/ wo 710   e. wcel 2176   {cab 2191   u. cun 3164  BOUNDED wbdc 15813

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 1470  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-4 1533  ax-17 1549  ax-ial 1557  ax-ext 2187  ax-bd0 15786  ax-bdor 15789  ax-bdsb 15795

This theorem depends on definitions:  df-bi 117  df-clab 2192  df-cleq 2198  df-clel 2201  df-un 3170  df-bdc 15814

This theorem is referenced by:  bdcpr  15844  bdctp  15845  bdcsuc  15853