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

Theorem nrelv 5813
Description: The universal class is not a relation. (Contributed by Thierry Arnoux, 23-Jan-2022.)
Assertion
Ref Expression
nrelv ¬ Rel V

Proof of Theorem nrelv
StepHypRef Expression
1 0ex 5313 . . 3 ∅ ∈ V
2 0nelxp 5723 . . 3 ¬ ∅ ∈ (V × V)
3 nelss 4061 . . 3 ((∅ ∈ V ∧ ¬ ∅ ∈ (V × V)) → ¬ V ⊆ (V × V))
41, 2, 3mp2an 692 . 2 ¬ V ⊆ (V × V)
5 df-rel 5696 . 2 (Rel V ↔ V ⊆ (V × V))
64, 5mtbir 323 1 ¬ Rel V
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wcel 2106  Vcvv 3478  wss 3963  c0 4339   × cxp 5687  Rel wrel 5694
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-opab 5211  df-xp 5695  df-rel 5696
This theorem is referenced by:  nfunv  6601  relintabex  43571
  Copyright terms: Public domain W3C validator