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

Theorem bdcint 11425
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 11369 . . . . 5  |- BOUNDED  y  e.  z
21ax-bdal 11366 . . . 4  |- BOUNDED  A. z  e.  x  y  e.  z
3 df-ral 2364 . . . 4  |-  ( A. z  e.  x  y  e.  z  <->  A. z ( z  e.  x  ->  y  e.  z ) )
42, 3bd0 11372 . . 3  |- BOUNDED  A. z ( z  e.  x  ->  y  e.  z )
54bdcab 11397 . 2  |- BOUNDED  { y  |  A. z ( z  e.  x  ->  y  e.  z ) }
6 df-int 3684 . 2  |-  |^| x  =  { y  |  A. z ( z  e.  x  ->  y  e.  z ) }
75, 6bdceqir 11392 1  |- BOUNDED 
|^| x
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1287   {cab 2074   A.wral 2359   |^|cint 3683  BOUNDED wbdc 11388
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-4 1445  ax-17 1464  ax-ial 1472  ax-ext 2070  ax-bd0 11361  ax-bdal 11366  ax-bdel 11369  ax-bdsb 11370
This theorem depends on definitions:  df-bi 115  df-clab 2075  df-cleq 2081  df-clel 2084  df-ral 2364  df-int 3684  df-bdc 11389
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator