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Theorem iun0 3839
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 3337 . . . . . 6 ¬ 𝑦 ∈ ∅
21a1i 9 . . . . 5 (𝑥𝐴 → ¬ 𝑦 ∈ ∅)
32nrex 2501 . . . 4 ¬ ∃𝑥𝐴 𝑦 ∈ ∅
4 eliun 3787 . . . 4 (𝑦 𝑥𝐴 ∅ ↔ ∃𝑥𝐴 𝑦 ∈ ∅)
53, 4mtbir 645 . . 3 ¬ 𝑦 𝑥𝐴
65, 12false 675 . 2 (𝑦 𝑥𝐴 ∅ ↔ 𝑦 ∈ ∅)
76eqriv 2114 1 𝑥𝐴 ∅ = ∅
Colors of variables: wff set class
Syntax hints:  ¬ wn 3   = wceq 1316  wcel 1465  wrex 2394  c0 3333   ciun 3783
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-dif 3043  df-nul 3334  df-iun 3785
This theorem is referenced by: (None)
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