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Theorem 0iun 4150
Description: An empty indexed union is empty. (Contributed by NM, 4-Dec-2004.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
Assertion
Ref Expression
0iun  |-  U_ x  e.  (/)  A  =  (/)

Proof of Theorem 0iun
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 rex0 3643 . . . 4  |-  -.  E. x  e.  (/)  y  e.  A
2 eliun 4099 . . . 4  |-  ( y  e.  U_ x  e.  (/)  A  <->  E. x  e.  (/)  y  e.  A )
31, 2mtbir 292 . . 3  |-  -.  y  e.  U_ x  e.  (/)  A
4 noel 3634 . . 3  |-  -.  y  e.  (/)
53, 42false 341 . 2  |-  ( y  e.  U_ x  e.  (/)  A  <->  y  e.  (/) )
65eqriv 2435 1  |-  U_ x  e.  (/)  A  =  (/)
Colors of variables: wff set class
Syntax hints:    = wceq 1653    e. wcel 1726   E.wrex 2708   (/)c0 3630   U_ciun 4095
This theorem is referenced by:  iununi  4177  iunfi  7396  pwsdompw  8086  fsum2d  12557  fsumiun  12602  prmreclem4  13289  prmreclem5  13290  fiuncmp  17469  ovolfiniun  19399  ovoliunnul  19405  finiunmbl  19440  volfiniun  19443  volsup  19452  fprod2d  25307  0totbnd  26484  totbndbnd  26500
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rex 2713  df-v 2960  df-dif 3325  df-nul 3631  df-iun 4097
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