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Theorem intexr 3895
 Description: If the intersection of a class exists, the class is non-empty. (Contributed by Jim Kingdon, 27-Aug-2018.)
Assertion
Ref Expression
intexr ( A V → A ≠ ∅)

Proof of Theorem intexr
StepHypRef Expression
1 vprc 3879 . . 3 ¬ V V
2 inteq 3609 . . . . 5 (A = ∅ → A = ∅)
3 int0 3620 . . . . 5 ∅ = V
42, 3syl6eq 2085 . . . 4 (A = ∅ → A = V)
54eleq1d 2103 . . 3 (A = ∅ → ( A V ↔ V V))
61, 5mtbiri 599 . 2 (A = ∅ → ¬ A V)
76necon2ai 2253 1 ( A V → A ≠ ∅)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1242   ∈ wcel 1390   ≠ wne 2201  Vcvv 2551  ∅c0 3218  ∩ cint 3606 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-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-fal 1248  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-int 3607 This theorem is referenced by: (None)
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