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Theorem iun0 3713
 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 3228 . . . . . 6 ¬ 𝑦 ∈ ∅
21a1i 9 . . . . 5 (𝑥𝐴 → ¬ 𝑦 ∈ ∅)
32nrex 2411 . . . 4 ¬ ∃𝑥𝐴 𝑦 ∈ ∅
4 eliun 3661 . . . 4 (𝑦 𝑥𝐴 ∅ ↔ ∃𝑥𝐴 𝑦 ∈ ∅)
53, 4mtbir 596 . . 3 ¬ 𝑦 𝑥𝐴
65, 12false 617 . 2 (𝑦 𝑥𝐴 ∅ ↔ 𝑦 ∈ ∅)
76eqriv 2037 1 𝑥𝐴 ∅ = ∅
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   = wceq 1243   ∈ wcel 1393  ∃wrex 2307  ∅c0 3224  ∪ ciun 3657 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-dif 2920  df-nul 3225  df-iun 3659 This theorem is referenced by: (None)
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