Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >   Mathboxes  >  bdcin Unicode version

Theorem bdcin 13898
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 13881 . . . 4  |- BOUNDED  x  e.  A
3 bdcdif.2 . . . . 5  |- BOUNDED  B
43bdeli 13881 . . . 4  |- BOUNDED  x  e.  B
52, 4ax-bdan 13850 . . 3  |- BOUNDED  ( x  e.  A  /\  x  e.  B
)
65bdcab 13884 . 2  |- BOUNDED  { x  |  ( x  e.  A  /\  x  e.  B ) }
7 df-in 3127 . 2  |-  ( A  i^i  B )  =  { x  |  ( x  e.  A  /\  x  e.  B ) }
86, 7bdceqir 13879 1  |- BOUNDED  ( A  i^i  B
)
Colors of variables: wff set class
Syntax hints:    /\ wa 103    e. wcel 2141   {cab 2156    i^i cin 3120  BOUNDED wbdc 13875
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 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503  ax-17 1519  ax-ial 1527  ax-ext 2152  ax-bd0 13848  ax-bdan 13850  ax-bdsb 13857
This theorem depends on definitions:  df-bi 116  df-clab 2157  df-cleq 2163  df-clel 2166  df-in 3127  df-bdc 13876
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator