Theorem nprrel 5683

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 5680	. 2 ⊢ (𝐴𝑅𝐵 → 𝐴 ∈ V)
4	1, 3	mto 197	1 ⊢ ¬ 𝐴𝑅𝐵

Syntax hints:  ¬ wn 3   ∈ wcel 2113  Vcvv 3440   class class class wbr 5098  Rel wrel 5629

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-br 5099  df-opab 5161  df-xp 5630  df-rel 5631

This theorem is referenced by: (None)