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Theorem nrelv 5471
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 5026 . . 3 ∅ ∈ V
2 0nelxp 5389 . . 3 ¬ ∅ ∈ (V × V)
3 nelss 3882 . . 3 ((∅ ∈ V ∧ ¬ ∅ ∈ (V × V)) → ¬ V ⊆ (V × V))
41, 2, 3mp2an 682 . 2 ¬ V ⊆ (V × V)
5 df-rel 5362 . 2 (Rel V ↔ V ⊆ (V × V))
64, 5mtbir 315 1 ¬ Rel V
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wcel 2106  Vcvv 3397  wss 3791  c0 4140   × cxp 5353  Rel wrel 5360
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-ext 2753  ax-sep 5017  ax-nul 5025  ax-pr 5138
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ne 2969  df-v 3399  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-opab 4949  df-xp 5361  df-rel 5362
This theorem is referenced by:  nfunv  6168
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