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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 4331 . . . . 5 ¬ 𝑦 ∈ ∅
21a1i 11 . . . 4 (𝑥𝐴 → ¬ 𝑦 ∈ ∅)
32nrex 3075 . . 3 ¬ ∃𝑥𝐴 𝑦 ∈ ∅
4 eliun 5002 . . 3 (𝑦 𝑥𝐴 ∅ ↔ ∃𝑥𝐴 𝑦 ∈ ∅)
53, 4mtbir 323 . 2 ¬ 𝑦 𝑥𝐴
65nel0 4351 1 𝑥𝐴 ∅ = ∅
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1542  wcel 2107  wrex 3071  c0 4323   ciun 4998
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-v 3477  df-dif 3952  df-nul 4324  df-iun 5000
This theorem is referenced by:  iunxdif3  5099  iununi  5103  funiunfv  7247  om0r  8539  kmlem11  10155  ituniiun  10417  dfrtrclrec2  15005  voliunlem1  25067  ofpreima2  31891  esum2dlem  33090  sigaclfu2  33119  measvunilem0  33211  measvuni  33212  cvmscld  34264  ovolval4lem1  45365
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