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Theorem nprrel 4631
Description: No proper class is related to anything via any relation. (Contributed by Roy F. Longton, 30-Jul-2005.)
Hypotheses
Ref Expression
nprrel.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 nprrel.1 . . 3 Rel 𝑅
32brrelex1i 4629 . 2 (𝐴𝑅𝐵𝐴 ∈ V)
41, 3mto 652 1 ¬ 𝐴𝑅𝐵
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wcel 2128  Vcvv 2712   class class class wbr 3965  Rel wrel 4591
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-br 3966  df-opab 4026  df-xp 4592  df-rel 4593
This theorem is referenced by: (None)
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