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Theorem bdcint 12041
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 11985 . . . . 5  |- BOUNDED  y  e.  z
21ax-bdal 11982 . . . 4  |- BOUNDED  A. z  e.  x  y  e.  z
3 df-ral 2365 . . . 4  |-  ( A. z  e.  x  y  e.  z  <->  A. z ( z  e.  x  ->  y  e.  z ) )
42, 3bd0 11988 . . 3  |- BOUNDED  A. z ( z  e.  x  ->  y  e.  z )
54bdcab 12013 . 2  |- BOUNDED  { y  |  A. z ( z  e.  x  ->  y  e.  z ) }
6 df-int 3695 . 2  |-  |^| x  =  { y  |  A. z ( z  e.  x  ->  y  e.  z ) }
75, 6bdceqir 12008 1  |- BOUNDED 
|^| x
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1288   {cab 2075   A.wral 2360   |^|cint 3694  BOUNDED wbdc 12004
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1382  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-4 1446  ax-17 1465  ax-ial 1473  ax-ext 2071  ax-bd0 11977  ax-bdal 11982  ax-bdel 11985  ax-bdsb 11986
This theorem depends on definitions:  df-bi 116  df-clab 2076  df-cleq 2082  df-clel 2085  df-ral 2365  df-int 3695  df-bdc 12005
This theorem is referenced by: (None)
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