Theorem nprrel12 5607

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 5604	. 2 ⊢ (𝐴𝑅𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
3	2	con3i 157	1 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ¬ 𝐴𝑅𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∈ wcel 2110  Vcvv 3408   class class class wbr 5053  Rel wrel 5556

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-br 5054  df-opab 5116  df-xp 5557  df-rel 5558

This theorem is referenced by: (None)