MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nprrel Structured version   Visualization version   GIF version

Theorem nprrel 5457
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
StepHypRef Expression
1 nprrel.2 . 2 ¬ 𝐴 ∈ V
2 nprrel12.1 . . 3 Rel 𝑅
32brrelex1i 5454 . 2 (𝐴𝑅𝐵𝐴 ∈ V)
41, 3mto 189 1 ¬ 𝐴𝑅𝐵
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wcel 2051  Vcvv 3408   class class class wbr 4925  Rel wrel 5408
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-ext 2743  ax-sep 5056  ax-nul 5063  ax-pr 5182
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-ral 3086  df-rex 3087  df-rab 3090  df-v 3410  df-dif 3825  df-un 3827  df-in 3829  df-ss 3836  df-nul 4173  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442  df-br 4926  df-opab 4988  df-xp 5409  df-rel 5410
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator