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Theorem iun0 3754
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 3271 . . . . . 6 ¬ 𝑦 ∈ ∅
21a1i 9 . . . . 5 (𝑥𝐴 → ¬ 𝑦 ∈ ∅)
32nrex 2458 . . . 4 ¬ ∃𝑥𝐴 𝑦 ∈ ∅
4 eliun 3702 . . . 4 (𝑦 𝑥𝐴 ∅ ↔ ∃𝑥𝐴 𝑦 ∈ ∅)
53, 4mtbir 629 . . 3 ¬ 𝑦 𝑥𝐴
65, 12false 650 . 2 (𝑦 𝑥𝐴 ∅ ↔ 𝑦 ∈ ∅)
76eqriv 2080 1 𝑥𝐴 ∅ = ∅
Colors of variables: wff set class
Syntax hints:  ¬ wn 3   = wceq 1285  wcel 1434  wrex 2354  c0 3267   ciun 3698
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 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2612  df-dif 2984  df-nul 3268  df-iun 3700
This theorem is referenced by: (None)
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