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Theorem nprrel12 5733
Description: Proper classes are not related via any relation. (Contributed by AV, 29-Oct-2021.)
Hypothesis
Ref Expression
nprrel12.1 Rel 𝑅
Assertion
Ref Expression
nprrel12 (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ¬ 𝐴𝑅𝐵)

Proof of Theorem nprrel12
StepHypRef Expression
1 nprrel12.1 . . 3 Rel 𝑅
21brrelex12i 5730 . 2 (𝐴𝑅𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
32con3i 154 1 (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ¬ 𝐴𝑅𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 397  wcel 2107  Vcvv 3475   class class class wbr 5148  Rel wrel 5681
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-xp 5682  df-rel 5683
This theorem is referenced by: (None)
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