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Theorem iin0r 3949
 Description: If an indexed intersection of the empty set is empty, the index set is non-empty. (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 3911 . . . . 5 ∅ ∈ V
2 n0i 3256 . . . . 5 (∅ ∈ V → ¬ V = ∅)
31, 2ax-mp 7 . . . 4 ¬ V = ∅
4 0iin 3742 . . . . 5 𝑥 ∈ ∅ ∅ = V
54eqeq1i 2063 . . . 4 ( 𝑥 ∈ ∅ ∅ = ∅ ↔ V = ∅)
63, 5mtbir 606 . . 3 ¬ 𝑥 ∈ ∅ ∅ = ∅
7 iineq1 3698 . . . 4 (𝐴 = ∅ → 𝑥𝐴 ∅ = 𝑥 ∈ ∅ ∅)
87eqeq1d 2064 . . 3 (𝐴 = ∅ → ( 𝑥𝐴 ∅ = ∅ ↔ 𝑥 ∈ ∅ ∅ = ∅))
96, 8mtbiri 610 . 2 (𝐴 = ∅ → ¬ 𝑥𝐴 ∅ = ∅)
109necon2ai 2274 1 ( 𝑥𝐴 ∅ = ∅ → 𝐴 ≠ ∅)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   = wceq 1259   ∈ wcel 1409   ≠ wne 2220  Vcvv 2574  ∅c0 3251  ∩ ciin 3685 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-nul 3910 This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-v 2576  df-dif 2947  df-nul 3252  df-iin 3687 This theorem is referenced by: (None)
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