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Theorem iin0r 3913
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 ( x A ∅ = ∅ → A ≠ ∅)
Distinct variable group:   x,A

Proof of Theorem iin0r
StepHypRef Expression
1 0ex 3875 . . . . 5 V
2 n0i 3223 . . . . 5 (∅ V → ¬ V = ∅)
31, 2ax-mp 7 . . . 4 ¬ V = ∅
4 0iin 3706 . . . . 5 x ∅ ∅ = V
54eqeq1i 2044 . . . 4 ( x ∅ ∅ = ∅ ↔ V = ∅)
63, 5mtbir 595 . . 3 ¬ x ∅ ∅ = ∅
7 iineq1 3662 . . . 4 (A = ∅ → x A ∅ = x ∅ ∅)
87eqeq1d 2045 . . 3 (A = ∅ → ( x A ∅ = ∅ ↔ x ∅ ∅ = ∅))
96, 8mtbiri 599 . 2 (A = ∅ → ¬ x A ∅ = ∅)
109necon2ai 2253 1 ( x A ∅ = ∅ → A ≠ ∅)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1242   wcel 1390  wne 2201  Vcvv 2551  c0 3218   ciin 3649
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-nul 3874
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-v 2553  df-dif 2914  df-nul 3219  df-iin 3651
This theorem is referenced by: (None)
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