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Theorem bdcin 13425
Description: The intersection 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
bdcin  |- BOUNDED  ( A  i^i  B
)

Proof of Theorem bdcin
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 bdcdif.1 . . . . 5  |- BOUNDED  A
21bdeli 13408 . . . 4  |- BOUNDED  x  e.  A
3 bdcdif.2 . . . . 5  |- BOUNDED  B
43bdeli 13408 . . . 4  |- BOUNDED  x  e.  B
52, 4ax-bdan 13377 . . 3  |- BOUNDED  ( x  e.  A  /\  x  e.  B
)
65bdcab 13411 . 2  |- BOUNDED  { x  |  ( x  e.  A  /\  x  e.  B ) }
7 df-in 3108 . 2  |-  ( A  i^i  B )  =  { x  |  ( x  e.  A  /\  x  e.  B ) }
86, 7bdceqir 13406 1  |- BOUNDED  ( A  i^i  B
)
Colors of variables: wff set class
Syntax hints:    /\ wa 103    e. wcel 2128   {cab 2143    i^i cin 3101  BOUNDED wbdc 13402
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 1427  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-4 1490  ax-17 1506  ax-ial 1514  ax-ext 2139  ax-bd0 13375  ax-bdan 13377  ax-bdsb 13384
This theorem depends on definitions:  df-bi 116  df-clab 2144  df-cleq 2150  df-clel 2153  df-in 3108  df-bdc 13403
This theorem is referenced by: (None)
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