Theorem nprrel12 5677

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

Step	Hyp	Ref	Expression
1		nprrel12.1	. . 3 ⊢ Rel 𝑅
2	1	brrelex12i 5674	. 2 ⊢ (𝐴𝑅𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
3	2	con3i 154	1 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ¬ 𝐴𝑅𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   ∈ wcel 2119  Vcvv 3431   class class class wbr 5073  Rel wrel 5624

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711  ax-sep 5219  ax-pr 5363

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4263  df-if 4456  df-sn 4557  df-pr 4559  df-op 4563  df-br 5074  df-opab 5136  df-xp 5625  df-rel 5626

This theorem is referenced by: (None)