ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  nprrel GIF version

Theorem nprrel 4656
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 4654 . 2 (𝐴𝑅𝐵𝐴 ∈ V)
41, 3mto 657 1 ¬ 𝐴𝑅𝐵
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wcel 2141  Vcvv 2730   class class class wbr 3989  Rel wrel 4616
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-xp 4617  df-rel 4618
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator