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Theorem bdcin 11400
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 11383 . . . 4  |- BOUNDED  x  e.  A
3 bdcdif.2 . . . . 5  |- BOUNDED  B
43bdeli 11383 . . . 4  |- BOUNDED  x  e.  B
52, 4ax-bdan 11352 . . 3  |- BOUNDED  ( x  e.  A  /\  x  e.  B
)
65bdcab 11386 . 2  |- BOUNDED  { x  |  ( x  e.  A  /\  x  e.  B ) }
7 df-in 3003 . 2  |-  ( A  i^i  B )  =  { x  |  ( x  e.  A  /\  x  e.  B ) }
86, 7bdceqir 11381 1  |- BOUNDED  ( A  i^i  B
)
Colors of variables: wff set class
Syntax hints:    /\ wa 102    e. wcel 1438   {cab 2074    i^i cin 2996  BOUNDED wbdc 11377
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-4 1445  ax-17 1464  ax-ial 1472  ax-ext 2070  ax-bd0 11350  ax-bdan 11352  ax-bdsb 11359
This theorem depends on definitions:  df-bi 115  df-clab 2075  df-cleq 2081  df-clel 2084  df-in 3003  df-bdc 11378
This theorem is referenced by: (None)
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