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Theorem iun0 3816
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 3314 . . . . . 6 ¬ 𝑦 ∈ ∅
21a1i 9 . . . . 5 (𝑥𝐴 → ¬ 𝑦 ∈ ∅)
32nrex 2483 . . . 4 ¬ ∃𝑥𝐴 𝑦 ∈ ∅
4 eliun 3764 . . . 4 (𝑦 𝑥𝐴 ∅ ↔ ∃𝑥𝐴 𝑦 ∈ ∅)
53, 4mtbir 637 . . 3 ¬ 𝑦 𝑥𝐴
65, 12false 658 . 2 (𝑦 𝑥𝐴 ∅ ↔ 𝑦 ∈ ∅)
76eqriv 2097 1 𝑥𝐴 ∅ = ∅
Colors of variables: wff set class
Syntax hints:  ¬ wn 3   = wceq 1299  wcel 1448  wrex 2376  c0 3310   ciun 3760
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082
This theorem depends on definitions:  df-bi 116  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-v 2643  df-dif 3023  df-nul 3311  df-iun 3762
This theorem is referenced by: (None)
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