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Theorem iun0 4982
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 4300 . . . . 5 ¬ 𝑦 ∈ ∅
21a1i 11 . . . 4 (𝑥𝐴 → ¬ 𝑦 ∈ ∅)
32nrex 3274 . . 3 ¬ ∃𝑥𝐴 𝑦 ∈ ∅
4 eliun 4921 . . 3 (𝑦 𝑥𝐴 ∅ ↔ ∃𝑥𝐴 𝑦 ∈ ∅)
53, 4mtbir 324 . 2 ¬ 𝑦 𝑥𝐴
65nel0 4315 1 𝑥𝐴 ∅ = ∅
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1530  wcel 2107  wrex 3144  c0 4295   ciun 4917
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ral 3148  df-rex 3149  df-v 3502  df-dif 3943  df-nul 4296  df-iun 4919
This theorem is referenced by:  iunxdif3  5014  iununi  5018  funiunfv  7001  om0r  8155  kmlem11  9575  ituniiun  9833  voliunlem1  24066  ofpreima2  30326  esum2dlem  31237  sigaclfu2  31266  measvunilem0  31358  measvuni  31359  cvmscld  32404  trpred0  32959  ovolval4lem1  42797
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