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Theorem iin0 4869
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 4562 . 2 (𝐴 ≠ ∅ → 𝑥𝐴 ∅ = ∅)
2 0ex 4823 . . . . . 6 ∅ ∈ V
32n0ii 3955 . . . . 5 ¬ V = ∅
4 0iin 4610 . . . . . 6 𝑥 ∈ ∅ ∅ = V
54eqeq1i 2656 . . . . 5 ( 𝑥 ∈ ∅ ∅ = ∅ ↔ V = ∅)
63, 5mtbir 312 . . . 4 ¬ 𝑥 ∈ ∅ ∅ = ∅
7 iineq1 4567 . . . . 5 (𝐴 = ∅ → 𝑥𝐴 ∅ = 𝑥 ∈ ∅ ∅)
87eqeq1d 2653 . . . 4 (𝐴 = ∅ → ( 𝑥𝐴 ∅ = ∅ ↔ 𝑥 ∈ ∅ ∅ = ∅))
96, 8mtbiri 316 . . 3 (𝐴 = ∅ → ¬ 𝑥𝐴 ∅ = ∅)
109necon2ai 2852 . 2 ( 𝑥𝐴 ∅ = ∅ → 𝐴 ≠ ∅)
111, 10impbii 199 1 (𝐴 ≠ ∅ ↔ 𝑥𝐴 ∅ = ∅)
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1523  wne 2823  Vcvv 3231  c0 3948   ciin 4553
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-nul 4822
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-v 3233  df-dif 3610  df-nul 3949  df-iin 4555
This theorem is referenced by: (None)
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