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Theorem intexr 4045
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 4030 . . 3 ¬ V ∈ V
2 inteq 3744 . . . . 5 (𝐴 = ∅ → 𝐴 = ∅)
3 int0 3755 . . . . 5 ∅ = V
42, 3syl6eq 2166 . . . 4 (𝐴 = ∅ → 𝐴 = V)
54eleq1d 2186 . . 3 (𝐴 = ∅ → ( 𝐴 ∈ V ↔ V ∈ V))
61, 5mtbiri 649 . 2 (𝐴 = ∅ → ¬ 𝐴 ∈ V)
76necon2ai 2339 1 ( 𝐴 ∈ V → 𝐴 ≠ ∅)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1316  wcel 1465  wne 2285  Vcvv 2660  c0 3333   cint 3741
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-v 2662  df-dif 3043  df-nul 3334  df-int 3742
This theorem is referenced by:  fival  6826
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