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Theorem iin0 5361
Description: An indexed intersection of the empty set, with a nonempty index set, is empty. (Contributed by NM, 20-Oct-2005.)
Assertion
Ref Expression
iin0 (𝐴 ≠ ∅ ↔ 𝑥𝐴 ∅ = ∅)
Distinct variable group:   𝑥,𝐴

Proof of Theorem iin0
StepHypRef Expression
1 iinconst 5008 . 2 (𝐴 ≠ ∅ → 𝑥𝐴 ∅ = ∅)
2 0ex 5308 . . . . . 6 ∅ ∈ V
32n0ii 4337 . . . . 5 ¬ V = ∅
4 0iin 5068 . . . . . 6 𝑥 ∈ ∅ ∅ = V
54eqeq1i 2738 . . . . 5 ( 𝑥 ∈ ∅ ∅ = ∅ ↔ V = ∅)
63, 5mtbir 323 . . . 4 ¬ 𝑥 ∈ ∅ ∅ = ∅
7 iineq1 5015 . . . . 5 (𝐴 = ∅ → 𝑥𝐴 ∅ = 𝑥 ∈ ∅ ∅)
87eqeq1d 2735 . . . 4 (𝐴 = ∅ → ( 𝑥𝐴 ∅ = ∅ ↔ 𝑥 ∈ ∅ ∅ = ∅))
96, 8mtbiri 327 . . 3 (𝐴 = ∅ → ¬ 𝑥𝐴 ∅ = ∅)
109necon2ai 2971 . 2 ( 𝑥𝐴 ∅ = ∅ → 𝐴 ≠ ∅)
111, 10impbii 208 1 (𝐴 ≠ ∅ ↔ 𝑥𝐴 ∅ = ∅)
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1542  wne 2941  Vcvv 3475  c0 4323   ciin 4999
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-12 2172  ax-ext 2704  ax-nul 5307
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-v 3477  df-dif 3952  df-nul 4324  df-iin 5001
This theorem is referenced by: (None)
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