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Theorem bdcin 10370
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 10353 . . . 4  |- BOUNDED  x  e.  A
3 bdcdif.2 . . . . 5  |- BOUNDED  B
43bdeli 10353 . . . 4  |- BOUNDED  x  e.  B
52, 4ax-bdan 10322 . . 3  |- BOUNDED  ( x  e.  A  /\  x  e.  B
)
65bdcab 10356 . 2  |- BOUNDED  { x  |  ( x  e.  A  /\  x  e.  B ) }
7 df-in 2952 . 2  |-  ( A  i^i  B )  =  { x  |  ( x  e.  A  /\  x  e.  B ) }
86, 7bdceqir 10351 1  |- BOUNDED  ( A  i^i  B
)
Colors of variables: wff set class
Syntax hints:    /\ wa 101    e. wcel 1409   {cab 2042    i^i cin 2944  BOUNDED wbdc 10347
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-4 1416  ax-17 1435  ax-ial 1443  ax-ext 2038  ax-bd0 10320  ax-bdan 10322  ax-bdsb 10329
This theorem depends on definitions:  df-bi 114  df-clab 2043  df-cleq 2049  df-clel 2052  df-in 2952  df-bdc 10348
This theorem is referenced by: (None)
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