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Theorem intexr 3901
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 3885 . . 3 ¬ V ∈ V
2 inteq 3615 . . . . 5 (𝐴 = ∅ → 𝐴 = ∅)
3 int0 3626 . . . . 5 ∅ = V
42, 3syl6eq 2088 . . . 4 (𝐴 = ∅ → 𝐴 = V)
54eleq1d 2106 . . 3 (𝐴 = ∅ → ( 𝐴 ∈ V ↔ V ∈ V))
61, 5mtbiri 600 . 2 (𝐴 = ∅ → ¬ 𝐴 ∈ V)
76necon2ai 2256 1 ( 𝐴 ∈ V → 𝐴 ≠ ∅)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1243  wcel 1393  wne 2204  Vcvv 2554  c0 3221   cint 3612
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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3872
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2308  df-v 2556  df-dif 2917  df-nul 3222  df-int 3613
This theorem is referenced by: (None)
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