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Theorem iin0 5252
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 4920 . 2 (𝐴 ≠ ∅ → 𝑥𝐴 ∅ = ∅)
2 0ex 5202 . . . . . 6 ∅ ∈ V
32n0ii 4299 . . . . 5 ¬ V = ∅
4 0iin 4978 . . . . . 6 𝑥 ∈ ∅ ∅ = V
54eqeq1i 2823 . . . . 5 ( 𝑥 ∈ ∅ ∅ = ∅ ↔ V = ∅)
63, 5mtbir 324 . . . 4 ¬ 𝑥 ∈ ∅ ∅ = ∅
7 iineq1 4927 . . . . 5 (𝐴 = ∅ → 𝑥𝐴 ∅ = 𝑥 ∈ ∅ ∅)
87eqeq1d 2820 . . . 4 (𝐴 = ∅ → ( 𝑥𝐴 ∅ = ∅ ↔ 𝑥 ∈ ∅ ∅ = ∅))
96, 8mtbiri 328 . . 3 (𝐴 = ∅ → ¬ 𝑥𝐴 ∅ = ∅)
109necon2ai 3042 . 2 ( 𝑥𝐴 ∅ = ∅ → 𝐴 ≠ ∅)
111, 10impbii 210 1 (𝐴 ≠ ∅ ↔ 𝑥𝐴 ∅ = ∅)
Colors of variables: wff setvar class
Syntax hints:  wb 207   = wceq 1528  wne 3013  Vcvv 3492  c0 4288   ciin 4911
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-nul 5201
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-v 3494  df-dif 3936  df-nul 4289  df-iin 4913
This theorem is referenced by: (None)
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