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Theorem nprrel12 5575
 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 5572 . 2 (𝐴𝑅𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
32con3i 157 1 (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ¬ 𝐴𝑅𝐵)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∈ wcel 2111  Vcvv 3441   class class class wbr 5031  Rel wrel 5525 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pr 5296 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-br 5032  df-opab 5094  df-xp 5526  df-rel 5527 This theorem is referenced by: (None)
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