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Theorem bdcuni 11197
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 11142 . . . . 5  |- BOUNDED  y  e.  z
21ax-bdex 11140 . . . 4  |- BOUNDED  E. z  e.  x  y  e.  z
32bdcab 11170 . . 3  |- BOUNDED  { y  |  E. z  e.  x  y  e.  z }
4 df-rex 2361 . . . . 5  |-  ( E. z  e.  x  y  e.  z  <->  E. z
( z  e.  x  /\  y  e.  z
) )
5 exancom 1542 . . . . 5  |-  ( E. z ( z  e.  x  /\  y  e.  z )  <->  E. z
( y  e.  z  /\  z  e.  x
) )
64, 5bitri 182 . . . 4  |-  ( E. z  e.  x  y  e.  z  <->  E. z
( y  e.  z  /\  z  e.  x
) )
76abbii 2200 . . 3  |-  { y  |  E. z  e.  x  y  e.  z }  =  { y  |  E. z ( y  e.  z  /\  z  e.  x ) }
83, 7bdceqi 11164 . 2  |- BOUNDED  { y  |  E. z ( y  e.  z  /\  z  e.  x ) }
9 df-uni 3637 . 2  |-  U. x  =  { y  |  E. z ( y  e.  z  /\  z  e.  x ) }
108, 9bdceqir 11165 1  |- BOUNDED 
U. x
Colors of variables: wff set class
Syntax hints:    /\ wa 102   E.wex 1424   {cab 2071   E.wrex 2356   U.cuni 3636  BOUNDED wbdc 11161
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 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-11 1440  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-bd0 11134  ax-bdex 11140  ax-bdel 11142  ax-bdsb 11143
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-rex 2361  df-uni 3637  df-bdc 11162
This theorem is referenced by:  bj-uniex2  11237
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