Theorem nprrel12 5194

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		brrelex12 5189	. . 3 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
3	1, 2	mpan 706	. 2 ⊢ (𝐴𝑅𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
4	3	con3i 150	1 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ¬ 𝐴𝑅𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   ∈ wcel 2030  Vcvv 3231   class class class wbr 4685  Rel wrel 5148

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-xp 5149  df-rel 5150

This theorem is referenced by: (None)