Theorem bdcun 15592

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 15576	. . . 4 |- BOUNDED x e. A
3		bdcdif.2	. . . . 5 |- BOUNDED B
4	3	bdeli 15576	. . . 4 |- BOUNDED x e. B
5	2, 4	ax-bdor 15546	. . 3 |- BOUNDED ( x e. A \/ x e. B )
6	5	bdcab 15579	. 2 |- BOUNDED { x | ( x e. A \/ x e. B ) }
7		df-un 3161	. 2 |- ( A u. B ) = { x | ( x e. A \/ x e. B ) }
8	6, 7	bdceqir 15574	1 |- BOUNDED ( A u. B )

Syntax hints:   \/ wo 709   e. wcel 2167   {cab 2182   u. cun 3155  BOUNDED wbdc 15570

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 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-4 1524  ax-17 1540  ax-ial 1548  ax-ext 2178  ax-bd0 15543  ax-bdor 15546  ax-bdsb 15552

This theorem depends on definitions:  df-bi 117  df-clab 2183  df-cleq 2189  df-clel 2192  df-un 3161  df-bdc 15571

This theorem is referenced by:  bdcpr  15601  bdctp  15602  bdcsuc  15610