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Theorem iun0 4732
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 4083 . . . . 5 ¬ 𝑦 ∈ ∅
21a1i 11 . . . 4 (𝑥𝐴 → ¬ 𝑦 ∈ ∅)
32nrex 3146 . . 3 ¬ ∃𝑥𝐴 𝑦 ∈ ∅
4 eliun 4680 . . 3 (𝑦 𝑥𝐴 ∅ ↔ ∃𝑥𝐴 𝑦 ∈ ∅)
53, 4mtbir 314 . 2 ¬ 𝑦 𝑥𝐴
65nel0 4096 1 𝑥𝐴 ∅ = ∅
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1652  wcel 2155  wrex 3056  c0 4079   ciun 4676
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-v 3352  df-dif 3735  df-nul 4080  df-iun 4678
This theorem is referenced by:  iunxdif3  4763  iununi  4767  funiunfv  6698  om0r  7824  kmlem11  9235  ituniiun  9497  voliunlem1  23608  ofpreima2  29851  esum2dlem  30536  sigaclfu2  30566  measvunilem0  30658  measvuni  30659  cvmscld  31635  trpred0  32111  ovolval4lem1  41435
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