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Theorem iun0 5009
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 4277 . . . . 5 ¬ 𝑦 ∈ ∅
21a1i 11 . . . 4 (𝑥𝐴 → ¬ 𝑦 ∈ ∅)
32nrex 3074 . . 3 ¬ ∃𝑥𝐴 𝑦 ∈ ∅
4 eliun 4945 . . 3 (𝑦 𝑥𝐴 ∅ ↔ ∃𝑥𝐴 𝑦 ∈ ∅)
53, 4mtbir 322 . 2 ¬ 𝑦 𝑥𝐴
65nel0 4297 1 𝑥𝐴 ∅ = ∅
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1540  wcel 2105  wrex 3070  c0 4269   ciun 4941
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3062  df-rex 3071  df-v 3443  df-dif 3901  df-nul 4270  df-iun 4943
This theorem is referenced by:  iunxdif3  5042  iununi  5046  funiunfv  7177  om0r  8440  kmlem11  10017  ituniiun  10279  dfrtrclrec2  14868  voliunlem1  24820  ofpreima2  31290  esum2dlem  32358  sigaclfu2  32387  measvunilem0  32479  measvuni  32480  cvmscld  33534  ovolval4lem1  44524
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