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Theorem bdcint 9997
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 9941 . . . . 5  |- BOUNDED  y  e.  z
21ax-bdal 9938 . . . 4  |- BOUNDED  A. z  e.  x  y  e.  z
3 df-ral 2311 . . . 4  |-  ( A. z  e.  x  y  e.  z  <->  A. z ( z  e.  x  ->  y  e.  z ) )
42, 3bd0 9944 . . 3  |- BOUNDED  A. z ( z  e.  x  ->  y  e.  z )
54bdcab 9969 . 2  |- BOUNDED  { y  |  A. z ( z  e.  x  ->  y  e.  z ) }
6 df-int 3616 . 2  |-  |^| x  =  { y  |  A. z ( z  e.  x  ->  y  e.  z ) }
75, 6bdceqir 9964 1  |- BOUNDED 
|^| x
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1241   {cab 2026   A.wral 2306   |^|cint 3615  BOUNDED wbdc 9960
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-17 1419  ax-ial 1427  ax-ext 2022  ax-bd0 9933  ax-bdal 9938  ax-bdel 9941  ax-bdsb 9942
This theorem depends on definitions:  df-bi 110  df-clab 2027  df-cleq 2033  df-clel 2036  df-ral 2311  df-int 3616  df-bdc 9961
This theorem is referenced by: (None)
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