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Theorem iun0 3958
Description: An indexed union of the empty set is empty. (Contributed by NM, 26-Mar-2003.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
Assertion
Ref Expression
iun0  |-  U_ x  e.  A  (/)  =  (/)

Proof of Theorem iun0
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 noel 3459 . . . . . 6  |-  -.  y  e.  (/)
21a1i 10 . . . . 5  |-  ( x  e.  A  ->  -.  y  e.  (/) )
32nrex 2645 . . . 4  |-  -.  E. x  e.  A  y  e.  (/)
4 eliun 3909 . . . 4  |-  ( y  e.  U_ x  e.  A  (/)  <->  E. x  e.  A  y  e.  (/) )
53, 4mtbir 290 . . 3  |-  -.  y  e.  U_ x  e.  A  (/)
65, 12false 339 . 2  |-  ( y  e.  U_ x  e.  A  (/)  <->  y  e.  (/) )
76eqriv 2280 1  |-  U_ x  e.  A  (/)  =  (/)
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1623    e. wcel 1684   E.wrex 2544   (/)c0 3455   U_ciun 3905
This theorem is referenced by:  iununi  3986  funiunfv  5774  om0r  6538  kmlem11  7786  ituniiun  8048  voliunlem1  18907  sigaclfu2  23482  measvunilem0  23541  measvuni  23542  cvmscld  23804  trpred0  24239
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-v 2790  df-dif 3155  df-nul 3456  df-iun 3907
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