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

Theorem bdcuni 14598
Description: The union of a setvar is a bounded class. (Contributed by BJ, 15-Oct-2019.)
Assertion
Ref Expression
bdcuni  |- BOUNDED 
U. x

Proof of Theorem bdcuni
Dummy variables  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ax-bdel 14543 . . . . 5  |- BOUNDED  y  e.  z
21ax-bdex 14541 . . . 4  |- BOUNDED  E. z  e.  x  y  e.  z
32bdcab 14571 . . 3  |- BOUNDED  { y  |  E. z  e.  x  y  e.  z }
4 df-rex 2461 . . . . 5  |-  ( E. z  e.  x  y  e.  z  <->  E. z
( z  e.  x  /\  y  e.  z
) )
5 exancom 1608 . . . . 5  |-  ( E. z ( z  e.  x  /\  y  e.  z )  <->  E. z
( y  e.  z  /\  z  e.  x
) )
64, 5bitri 184 . . . 4  |-  ( E. z  e.  x  y  e.  z  <->  E. z
( y  e.  z  /\  z  e.  x
) )
76abbii 2293 . . 3  |-  { y  |  E. z  e.  x  y  e.  z }  =  { y  |  E. z ( y  e.  z  /\  z  e.  x ) }
83, 7bdceqi 14565 . 2  |- BOUNDED  { y  |  E. z ( y  e.  z  /\  z  e.  x ) }
9 df-uni 3810 . 2  |-  U. x  =  { y  |  E. z ( y  e.  z  /\  z  e.  x ) }
108, 9bdceqir 14566 1  |- BOUNDED 
U. x
Colors of variables: wff set class
Syntax hints:    /\ wa 104   E.wex 1492   {cab 2163   E.wrex 2456   U.cuni 3809  BOUNDED wbdc 14562
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159  ax-bd0 14535  ax-bdex 14541  ax-bdel 14543  ax-bdsb 14544
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-rex 2461  df-uni 3810  df-bdc 14563
This theorem is referenced by:  bj-uniex2  14638
  Copyright terms: Public domain W3C validator