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Theorem iin0r 4153
Description: If an indexed intersection of the empty set is empty, the index set is nonempty. (Contributed by Jim Kingdon, 29-Aug-2018.)
Assertion
Ref Expression
iin0r ( 𝑥𝐴 ∅ = ∅ → 𝐴 ≠ ∅)
Distinct variable group:   𝑥,𝐴

Proof of Theorem iin0r
StepHypRef Expression
1 0ex 4114 . . . . 5 ∅ ∈ V
2 n0i 3419 . . . . 5 (∅ ∈ V → ¬ V = ∅)
31, 2ax-mp 5 . . . 4 ¬ V = ∅
4 0iin 3929 . . . . 5 𝑥 ∈ ∅ ∅ = V
54eqeq1i 2178 . . . 4 ( 𝑥 ∈ ∅ ∅ = ∅ ↔ V = ∅)
63, 5mtbir 666 . . 3 ¬ 𝑥 ∈ ∅ ∅ = ∅
7 iineq1 3885 . . . 4 (𝐴 = ∅ → 𝑥𝐴 ∅ = 𝑥 ∈ ∅ ∅)
87eqeq1d 2179 . . 3 (𝐴 = ∅ → ( 𝑥𝐴 ∅ = ∅ ↔ 𝑥 ∈ ∅ ∅ = ∅))
96, 8mtbiri 670 . 2 (𝐴 = ∅ → ¬ 𝑥𝐴 ∅ = ∅)
109necon2ai 2394 1 ( 𝑥𝐴 ∅ = ∅ → 𝐴 ≠ ∅)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1348  wcel 2141  wne 2340  Vcvv 2730  c0 3414   ciin 3872
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152  ax-nul 4113
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-v 2732  df-dif 3123  df-nul 3415  df-iin 3874
This theorem is referenced by: (None)
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