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Theorem nrelv 5672
 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 5210 . . 3 ∅ ∈ V
2 0nelxp 5588 . . 3 ¬ ∅ ∈ (V × V)
3 nelss 4029 . . 3 ((∅ ∈ V ∧ ¬ ∅ ∈ (V × V)) → ¬ V ⊆ (V × V))
41, 2, 3mp2an 690 . 2 ¬ V ⊆ (V × V)
5 df-rel 5561 . 2 (Rel V ↔ V ⊆ (V × V))
64, 5mtbir 325 1 ¬ Rel V
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∈ wcel 2110  Vcvv 3494   ⊆ wss 3935  ∅c0 4290   × cxp 5552  Rel wrel 5559 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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pr 5329 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4567  df-pr 4569  df-op 4573  df-opab 5128  df-xp 5560  df-rel 5561 This theorem is referenced by:  nfunv  6387  relintabex  39939
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