Theorem nprrel 5748

Description: No proper class is related to anything via any relation. (Contributed by Roy F. Longton, 30-Jul-2005.)

Hypotheses

Ref	Expression
nprrel12.1	⊢ Rel 𝑅
nprrel.2	⊢ ¬ 𝐴 ∈ V

Assertion

Ref	Expression
nprrel	⊢ ¬ 𝐴𝑅𝐵

Proof of Theorem nprrel

Step	Hyp	Ref	Expression
1		nprrel.2	. 2 ⊢ ¬ 𝐴 ∈ V
2		nprrel12.1	. . 3 ⊢ Rel 𝑅
3	2	brrelex1i 5745	. 2 ⊢ (𝐴𝑅𝐵 → 𝐴 ∈ V)
4	1, 3	mto 197	1 ⊢ ¬ 𝐴𝑅𝐵

Syntax hints:  ¬ wn 3   ∈ wcel 2106  Vcvv 3478   class class class wbr 5148  Rel wrel 5694

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696

This theorem is referenced by: (None)