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Theorem nrelv 5810
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 5307 . . 3 ∅ ∈ V
2 0nelxp 5719 . . 3 ¬ ∅ ∈ (V × V)
3 nelss 4049 . . 3 ((∅ ∈ V ∧ ¬ ∅ ∈ (V × V)) → ¬ V ⊆ (V × V))
41, 2, 3mp2an 692 . 2 ¬ V ⊆ (V × V)
5 df-rel 5692 . 2 (Rel V ↔ V ⊆ (V × V))
64, 5mtbir 323 1 ¬ Rel V
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wcel 2108  Vcvv 3480  wss 3951  c0 4333   × cxp 5683  Rel wrel 5690
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-opab 5206  df-xp 5691  df-rel 5692
This theorem is referenced by:  nfunv  6599  relintabex  43594
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