Theorem nprrel 5737

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 5734	. 2 ⊢ (𝐴𝑅𝐵 → 𝐴 ∈ V)
4	1, 3	mto 196	1 ⊢ ¬ 𝐴𝑅𝐵

Syntax hints:  ¬ wn 3   ∈ wcel 2099  Vcvv 3471   class class class wbr 5148  Rel wrel 5683

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5149  df-opab 5211  df-xp 5684  df-rel 5685

This theorem is referenced by: (None)