Theorem bdcun 16888

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 16872	. . . 4 |- BOUNDED x e. A
3		bdcdif.2	. . . . 5 |- BOUNDED B
4	3	bdeli 16872	. . . 4 |- BOUNDED x e. B
5	2, 4	ax-bdor 16842	. . 3 |- BOUNDED ( x e. A \/ x e. B )
6	5	bdcab 16875	. 2 |- BOUNDED { x | ( x e. A \/ x e. B ) }
7		df-un 3224	. 2 |- ( A u. B ) = { x | ( x e. A \/ x e. B ) }
8	6, 7	bdceqir 16870	1 |- BOUNDED ( A u. B )

This proof depends on syntax axioms:   \/ wo 720   e. wcel 2209   {cab 2224   u. cun 3218  BOUNDED wbdc 16866

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-4 1563  ax-17 1579  ax-ial 1587  ax-ext 2220  ax-bd0 16839  ax-bdor 16842  ax-bdsb 16848

This proof depends on definitions:  df-bi 117  df-clab 2225  df-cleq 2231  df-clel 2234  df-un 3224  df-bdc 16867

This theorem is used by:  bdcpr  16897  bdctp  16898  bdcsuc  16906