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Theorem iin0 4804
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 4501 . 2 (𝐴 ≠ ∅ → 𝑥𝐴 ∅ = ∅)
2 0ex 4755 . . . . . 6 ∅ ∈ V
32n0ii 3903 . . . . 5 ¬ V = ∅
4 0iin 4549 . . . . . 6 𝑥 ∈ ∅ ∅ = V
54eqeq1i 2626 . . . . 5 ( 𝑥 ∈ ∅ ∅ = ∅ ↔ V = ∅)
63, 5mtbir 313 . . . 4 ¬ 𝑥 ∈ ∅ ∅ = ∅
7 iineq1 4506 . . . . 5 (𝐴 = ∅ → 𝑥𝐴 ∅ = 𝑥 ∈ ∅ ∅)
87eqeq1d 2623 . . . 4 (𝐴 = ∅ → ( 𝑥𝐴 ∅ = ∅ ↔ 𝑥 ∈ ∅ ∅ = ∅))
96, 8mtbiri 317 . . 3 (𝐴 = ∅ → ¬ 𝑥𝐴 ∅ = ∅)
109necon2ai 2819 . 2 ( 𝑥𝐴 ∅ = ∅ → 𝐴 ≠ ∅)
111, 10impbii 199 1 (𝐴 ≠ ∅ ↔ 𝑥𝐴 ∅ = ∅)
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1480  wne 2790  Vcvv 3189  c0 3896   ciin 4491
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-nul 4754
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-v 3191  df-dif 3562  df-nul 3897  df-iin 4493
This theorem is referenced by: (None)
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