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Theorem bdcuni 12876
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 12821 . . . . 5  |- BOUNDED  y  e.  z
21ax-bdex 12819 . . . 4  |- BOUNDED  E. z  e.  x  y  e.  z
32bdcab 12849 . . 3  |- BOUNDED  { y  |  E. z  e.  x  y  e.  z }
4 df-rex 2397 . . . . 5  |-  ( E. z  e.  x  y  e.  z  <->  E. z
( z  e.  x  /\  y  e.  z
) )
5 exancom 1570 . . . . 5  |-  ( E. z ( z  e.  x  /\  y  e.  z )  <->  E. z
( y  e.  z  /\  z  e.  x
) )
64, 5bitri 183 . . . 4  |-  ( E. z  e.  x  y  e.  z  <->  E. z
( y  e.  z  /\  z  e.  x
) )
76abbii 2231 . . 3  |-  { y  |  E. z  e.  x  y  e.  z }  =  { y  |  E. z ( y  e.  z  /\  z  e.  x ) }
83, 7bdceqi 12843 . 2  |- BOUNDED  { y  |  E. z ( y  e.  z  /\  z  e.  x ) }
9 df-uni 3705 . 2  |-  U. x  =  { y  |  E. z ( y  e.  z  /\  z  e.  x ) }
108, 9bdceqir 12844 1  |- BOUNDED 
U. x
Colors of variables: wff set class
Syntax hints:    /\ wa 103   E.wex 1451   {cab 2101   E.wrex 2392   U.cuni 3704  BOUNDED wbdc 12840
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 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-11 1467  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-bd0 12813  ax-bdex 12819  ax-bdel 12821  ax-bdsb 12822
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-rex 2397  df-uni 3705  df-bdc 12841
This theorem is referenced by:  bj-uniex2  12916
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