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Theorem iun0 4374
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 3782 . . . . . 6  |-  -.  y  e.  (/)
21a1i 11 . . . . 5  |-  ( x  e.  A  ->  -.  y  e.  (/) )
32nrex 2912 . . . 4  |-  -.  E. x  e.  A  y  e.  (/)
4 eliun 4323 . . . 4  |-  ( y  e.  U_ x  e.  A  (/)  <->  E. x  e.  A  y  e.  (/) )
53, 4mtbir 299 . . 3  |-  -.  y  e.  U_ x  e.  A  (/)
65, 12false 350 . 2  |-  ( y  e.  U_ x  e.  A  (/)  <->  y  e.  (/) )
76eqriv 2456 1  |-  U_ x  e.  A  (/)  =  (/)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1374    e. wcel 1762   E.wrex 2808   (/)c0 3778   U_ciun 4318
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ral 2812  df-rex 2813  df-v 3108  df-dif 3472  df-nul 3779  df-iun 4320
This theorem is referenced by:  iununi  4403  funiunfv  6139  om0r  7179  kmlem11  8529  ituniiun  8791  voliunlem1  21688  ofpreima2  27030  sigaclfu2  27611  measvunilem0  27674  measvuni  27675  cvmscld  28208  trpred0  28746
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