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

Theorem bdcint 13134
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 13078 . . . . 5  |- BOUNDED  y  e.  z
21ax-bdal 13075 . . . 4  |- BOUNDED  A. z  e.  x  y  e.  z
3 df-ral 2421 . . . 4  |-  ( A. z  e.  x  y  e.  z  <->  A. z ( z  e.  x  ->  y  e.  z ) )
42, 3bd0 13081 . . 3  |- BOUNDED  A. z ( z  e.  x  ->  y  e.  z )
54bdcab 13106 . 2  |- BOUNDED  { y  |  A. z ( z  e.  x  ->  y  e.  z ) }
6 df-int 3772 . 2  |-  |^| x  =  { y  |  A. z ( z  e.  x  ->  y  e.  z ) }
75, 6bdceqir 13101 1  |- BOUNDED 
|^| x
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1329   {cab 2125   A.wral 2416   |^|cint 3771  BOUNDED wbdc 13097
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 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-17 1506  ax-ial 1514  ax-ext 2121  ax-bd0 13070  ax-bdal 13075  ax-bdel 13078  ax-bdsb 13079
This theorem depends on definitions:  df-bi 116  df-clab 2126  df-cleq 2132  df-clel 2135  df-ral 2421  df-int 3772  df-bdc 13098
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator