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Theorem nrelv 5644
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 5175 . . 3 ∅ ∈ V
2 0nelxp 5559 . . 3 ¬ ∅ ∈ (V × V)
3 nelss 3940 . . 3 ((∅ ∈ V ∧ ¬ ∅ ∈ (V × V)) → ¬ V ⊆ (V × V))
41, 2, 3mp2an 692 . 2 ¬ V ⊆ (V × V)
5 df-rel 5532 . 2 (Rel V ↔ V ⊆ (V × V))
64, 5mtbir 326 1 ¬ Rel V
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wcel 2114  Vcvv 3398  wss 3843  c0 4211   × cxp 5523  Rel wrel 5530
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pr 5296
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2075  df-clab 2717  df-cleq 2730  df-clel 2811  df-ne 2935  df-v 3400  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-sn 4517  df-pr 4519  df-op 4523  df-opab 5093  df-xp 5531  df-rel 5532
This theorem is referenced by:  nfunv  6372  relintabex  40734
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