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Theorem iun0 4710
 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 4067 . . . . 5 ¬ 𝑦 ∈ ∅
21a1i 11 . . . 4 (𝑥𝐴 → ¬ 𝑦 ∈ ∅)
32nrex 3148 . . 3 ¬ ∃𝑥𝐴 𝑦 ∈ ∅
4 eliun 4658 . . 3 (𝑦 𝑥𝐴 ∅ ↔ ∃𝑥𝐴 𝑦 ∈ ∅)
53, 4mtbir 312 . 2 ¬ 𝑦 𝑥𝐴
65nel0 4079 1 𝑥𝐴 ∅ = ∅
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   = wceq 1631   ∈ wcel 2145  ∃wrex 3062  ∅c0 4063  ∪ ciun 4654 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-v 3353  df-dif 3726  df-nul 4064  df-iun 4656 This theorem is referenced by:  iunxdif3  4740  iununi  4744  funiunfv  6649  om0r  7773  kmlem11  9184  ituniiun  9446  voliunlem1  23538  ofpreima2  29806  esum2dlem  30494  sigaclfu2  30524  measvunilem0  30616  measvuni  30617  cvmscld  31593  trpred0  32072  ovolval4lem1  41383
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