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Theorem iun0 3771
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 3279 . . . . . 6 ¬ 𝑦 ∈ ∅
21a1i 9 . . . . 5 (𝑥𝐴 → ¬ 𝑦 ∈ ∅)
32nrex 2461 . . . 4 ¬ ∃𝑥𝐴 𝑦 ∈ ∅
4 eliun 3719 . . . 4 (𝑦 𝑥𝐴 ∅ ↔ ∃𝑥𝐴 𝑦 ∈ ∅)
53, 4mtbir 629 . . 3 ¬ 𝑦 𝑥𝐴
65, 12false 650 . 2 (𝑦 𝑥𝐴 ∅ ↔ 𝑦 ∈ ∅)
76eqriv 2082 1 𝑥𝐴 ∅ = ∅
Colors of variables: wff set class
Syntax hints:  ¬ wn 3   = wceq 1287  wcel 1436  wrex 2356  c0 3275   ciun 3715
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-v 2617  df-dif 2990  df-nul 3276  df-iun 3717
This theorem is referenced by: (None)
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