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Theorem bdcuni 10383
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 10328 . . . . 5  |- BOUNDED  y  e.  z
21ax-bdex 10326 . . . 4  |- BOUNDED  E. z  e.  x  y  e.  z
32bdcab 10356 . . 3  |- BOUNDED  { y  |  E. z  e.  x  y  e.  z }
4 df-rex 2329 . . . . 5  |-  ( E. z  e.  x  y  e.  z  <->  E. z
( z  e.  x  /\  y  e.  z
) )
5 exancom 1515 . . . . 5  |-  ( E. z ( z  e.  x  /\  y  e.  z )  <->  E. z
( y  e.  z  /\  z  e.  x
) )
64, 5bitri 177 . . . 4  |-  ( E. z  e.  x  y  e.  z  <->  E. z
( y  e.  z  /\  z  e.  x
) )
76abbii 2169 . . 3  |-  { y  |  E. z  e.  x  y  e.  z }  =  { y  |  E. z ( y  e.  z  /\  z  e.  x ) }
83, 7bdceqi 10350 . 2  |- BOUNDED  { y  |  E. z ( y  e.  z  /\  z  e.  x ) }
9 df-uni 3609 . 2  |-  U. x  =  { y  |  E. z ( y  e.  z  /\  z  e.  x ) }
108, 9bdceqir 10351 1  |- BOUNDED 
U. x
Colors of variables: wff set class
Syntax hints:    /\ wa 101   E.wex 1397   {cab 2042   E.wrex 2324   U.cuni 3608  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-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-11 1413  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-bd0 10320  ax-bdex 10326  ax-bdel 10328  ax-bdsb 10329
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-rex 2329  df-uni 3609  df-bdc 10348
This theorem is referenced by:  bj-uniex2  10423
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