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Theorem bdcint 10384
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 10328 . . . . 5  |- BOUNDED  y  e.  z
21ax-bdal 10325 . . . 4  |- BOUNDED  A. z  e.  x  y  e.  z
3 df-ral 2328 . . . 4  |-  ( A. z  e.  x  y  e.  z  <->  A. z ( z  e.  x  ->  y  e.  z ) )
42, 3bd0 10331 . . 3  |- BOUNDED  A. z ( z  e.  x  ->  y  e.  z )
54bdcab 10356 . 2  |- BOUNDED  { y  |  A. z ( z  e.  x  ->  y  e.  z ) }
6 df-int 3644 . 2  |-  |^| x  =  { y  |  A. z ( z  e.  x  ->  y  e.  z ) }
75, 6bdceqir 10351 1  |- BOUNDED 
|^| x
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1257   {cab 2042   A.wral 2323   |^|cint 3643  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-bdal 10325  ax-bdel 10328  ax-bdsb 10329
This theorem depends on definitions:  df-bi 114  df-clab 2043  df-cleq 2049  df-clel 2052  df-ral 2328  df-int 3644  df-bdc 10348
This theorem is referenced by: (None)
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