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Theorem iun0 4512
 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 3878 . . . . . 6 ¬ 𝑦 ∈ ∅
21a1i 11 . . . . 5 (𝑥𝐴 → ¬ 𝑦 ∈ ∅)
32nrex 2983 . . . 4 ¬ ∃𝑥𝐴 𝑦 ∈ ∅
4 eliun 4460 . . . 4 (𝑦 𝑥𝐴 ∅ ↔ ∃𝑥𝐴 𝑦 ∈ ∅)
53, 4mtbir 312 . . 3 ¬ 𝑦 𝑥𝐴
65, 12false 364 . 2 (𝑦 𝑥𝐴 ∅ ↔ 𝑦 ∈ ∅)
76eqriv 2607 1 𝑥𝐴 ∅ = ∅
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   = wceq 1475   ∈ wcel 1977  ∃wrex 2897  ∅c0 3874  ∪ ciun 4455 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-v 3175  df-dif 3543  df-nul 3875  df-iun 4457 This theorem is referenced by:  iunxdif3  4542  iununi  4546  funiunfv  6410  om0r  7506  kmlem11  8865  ituniiun  9127  voliunlem1  23125  ofpreima2  28849  esum2dlem  29481  sigaclfu2  29511  measvunilem0  29603  measvuni  29604  cvmscld  30509  trpred0  30980  ovolval4lem1  39539
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