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Theorem iun0 5066
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 𝑥𝐴 ∅ = ∅

Proof of Theorem iun0
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 noel 4343 . . . . 5 ¬ 𝑦 ∈ ∅
21a1i 11 . . . 4 (𝑥𝐴 → ¬ 𝑦 ∈ ∅)
32nrex 3071 . . 3 ¬ ∃𝑥𝐴 𝑦 ∈ ∅
4 eliun 4999 . . 3 (𝑦 𝑥𝐴 ∅ ↔ ∃𝑥𝐴 𝑦 ∈ ∅)
53, 4mtbir 323 . 2 ¬ 𝑦 𝑥𝐴
65nel0 4359 1 𝑥𝐴 ∅ = ∅
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1536  wcel 2105  wrex 3067  c0 4338   ciun 4995
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ral 3059  df-rex 3068  df-v 3479  df-dif 3965  df-nul 4339  df-iun 4997
This theorem is referenced by:  iunxdif3  5099  iununi  5103  funiunfv  7267  om0r  8575  kmlem11  10198  ituniiun  10459  dfrtrclrec2  15093  voliunlem1  25598  ofpreima2  32682  ssdifidllem  33463  esum2dlem  34072  sigaclfu2  34101  measvunilem0  34193  measvuni  34194  cvmscld  35257  ovolval4lem1  46604
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