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Theorem 0iun 4382
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 3799 . . . 4  |-  -.  E. x  e.  (/)  y  e.  A
2 eliun 4330 . . . 4  |-  ( y  e.  U_ x  e.  (/)  A  <->  E. x  e.  (/)  y  e.  A )
31, 2mtbir 299 . . 3  |-  -.  y  e.  U_ x  e.  (/)  A
4 noel 3789 . . 3  |-  -.  y  e.  (/)
53, 42false 350 . 2  |-  ( y  e.  U_ x  e.  (/)  A  <->  y  e.  (/) )
65eqriv 2463 1  |-  U_ x  e.  (/)  A  =  (/)
Colors of variables: wff setvar class
Syntax hints:    = wceq 1379    e. wcel 1767   E.wrex 2815   (/)c0 3785   U_ciun 4325
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2819  df-rex 2820  df-v 3115  df-dif 3479  df-nul 3786  df-iun 4327
This theorem is referenced by:  iinvdif  4397  iununi  4410  iunfi  7808  pwsdompw  8584  fsum2d  13549  fsumiun  13598  prmreclem4  14296  prmreclem5  14297  fiuncmp  19698  ovolfiniun  21675  ovoliunnul  21681  finiunmbl  21717  volfiniun  21720  volsup  21729  fprod2d  28716  0totbnd  29900  totbndbnd  29916
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