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Theorem nprrel 5579
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 5576 . 2 (𝐴𝑅𝐵𝐴 ∈ V)
41, 3mto 200 1 ¬ 𝐴𝑅𝐵
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wcel 2112  Vcvv 3444   class class class wbr 5033  Rel wrel 5528
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-v 3446  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-br 5034  df-opab 5096  df-xp 5529  df-rel 5530
This theorem is referenced by: (None)
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