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Theorem intexr 4015
Description: If the intersection of a class exists, the class is nonempty. (Contributed by Jim Kingdon, 27-Aug-2018.)
Assertion
Ref Expression
intexr ( 𝐴 ∈ V → 𝐴 ≠ ∅)

Proof of Theorem intexr
StepHypRef Expression
1 vprc 4000 . . 3 ¬ V ∈ V
2 inteq 3721 . . . . 5 (𝐴 = ∅ → 𝐴 = ∅)
3 int0 3732 . . . . 5 ∅ = V
42, 3syl6eq 2148 . . . 4 (𝐴 = ∅ → 𝐴 = V)
54eleq1d 2168 . . 3 (𝐴 = ∅ → ( 𝐴 ∈ V ↔ V ∈ V))
61, 5mtbiri 641 . 2 (𝐴 = ∅ → ¬ 𝐴 ∈ V)
76necon2ai 2321 1 ( 𝐴 ∈ V → 𝐴 ≠ ∅)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1299  wcel 1448  wne 2267  Vcvv 2641  c0 3310   cint 3718
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986
This theorem depends on definitions:  df-bi 116  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-ral 2380  df-v 2643  df-dif 3023  df-nul 3311  df-int 3719
This theorem is referenced by: (None)
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