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Theorem bdcint 14925
Description: The intersection of a setvar is a bounded class. (Contributed by BJ, 16-Oct-2019.)
Assertion
Ref Expression
bdcint  |- BOUNDED 
|^| x

Proof of Theorem bdcint
Dummy variables  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ax-bdel 14869 . . . . 5  |- BOUNDED  y  e.  z
21ax-bdal 14866 . . . 4  |- BOUNDED  A. z  e.  x  y  e.  z
3 df-ral 2470 . . . 4  |-  ( A. z  e.  x  y  e.  z  <->  A. z ( z  e.  x  ->  y  e.  z ) )
42, 3bd0 14872 . . 3  |- BOUNDED  A. z ( z  e.  x  ->  y  e.  z )
54bdcab 14897 . 2  |- BOUNDED  { y  |  A. z ( z  e.  x  ->  y  e.  z ) }
6 df-int 3857 . 2  |-  |^| x  =  { y  |  A. z ( z  e.  x  ->  y  e.  z ) }
75, 6bdceqir 14892 1  |- BOUNDED 
|^| x
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1361   {cab 2173   A.wral 2465   |^|cint 3856  BOUNDED wbdc 14888
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1457  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-4 1520  ax-17 1536  ax-ial 1544  ax-ext 2169  ax-bd0 14861  ax-bdal 14866  ax-bdel 14869  ax-bdsb 14870
This theorem depends on definitions:  df-bi 117  df-clab 2174  df-cleq 2180  df-clel 2183  df-ral 2470  df-int 3857  df-bdc 14889
This theorem is referenced by: (None)
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