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Theorem iin0r 3972
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 3934 . . . . 5 ∅ ∈ V
2 n0i 3277 . . . . 5 (∅ ∈ V → ¬ V = ∅)
31, 2ax-mp 7 . . . 4 ¬ V = ∅
4 0iin 3765 . . . . 5 𝑥 ∈ ∅ ∅ = V
54eqeq1i 2092 . . . 4 ( 𝑥 ∈ ∅ ∅ = ∅ ↔ V = ∅)
63, 5mtbir 629 . . 3 ¬ 𝑥 ∈ ∅ ∅ = ∅
7 iineq1 3721 . . . 4 (𝐴 = ∅ → 𝑥𝐴 ∅ = 𝑥 ∈ ∅ ∅)
87eqeq1d 2093 . . 3 (𝐴 = ∅ → ( 𝑥𝐴 ∅ = ∅ ↔ 𝑥 ∈ ∅ ∅ = ∅))
96, 8mtbiri 633 . 2 (𝐴 = ∅ → ¬ 𝑥𝐴 ∅ = ∅)
109necon2ai 2305 1 ( 𝑥𝐴 ∅ = ∅ → 𝐴 ≠ ∅)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1287  wcel 1436  wne 2251  Vcvv 2614  c0 3272   ciin 3708
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-nul 3933
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-ral 2360  df-v 2616  df-dif 2988  df-nul 3273  df-iin 3710
This theorem is referenced by: (None)
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