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Theorem bdcun 13204
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.
StepHypRef Expression
1 bdcdif.1 . . . . 5  |- BOUNDED  A
21bdeli 13188 . . . 4  |- BOUNDED  x  e.  A
3 bdcdif.2 . . . . 5  |- BOUNDED  B
43bdeli 13188 . . . 4  |- BOUNDED  x  e.  B
52, 4ax-bdor 13158 . . 3  |- BOUNDED  ( x  e.  A  \/  x  e.  B
)
65bdcab 13191 . 2  |- BOUNDED  { x  |  ( x  e.  A  \/  x  e.  B ) }
7 df-un 3075 . 2  |-  ( A  u.  B )  =  { x  |  ( x  e.  A  \/  x  e.  B ) }
86, 7bdceqir 13186 1  |- BOUNDED  ( A  u.  B
)
Colors of variables: wff set class
Syntax hints:    \/ wo 697    e. wcel 1480   {cab 2125    u. cun 3069  BOUNDED wbdc 13182
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 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-17 1506  ax-ial 1514  ax-ext 2121  ax-bd0 13155  ax-bdor 13158  ax-bdsb 13164
This theorem depends on definitions:  df-bi 116  df-clab 2126  df-cleq 2132  df-clel 2135  df-un 3075  df-bdc 13183
This theorem is referenced by:  bdcpr  13213  bdctp  13214  bdcsuc  13222
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