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Theorem nrelv 5400
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 4942 . . 3 ∅ ∈ V
2 0nelxp 5300 . . 3 ¬ ∅ ∈ (V × V)
3 nelss 3805 . . 3 ((∅ ∈ V ∧ ¬ ∅ ∈ (V × V)) → ¬ V ⊆ (V × V))
41, 2, 3mp2an 710 . 2 ¬ V ⊆ (V × V)
5 df-rel 5273 . 2 (Rel V ↔ V ⊆ (V × V))
64, 5mtbir 312 1 ¬ Rel V
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wcel 2139  Vcvv 3340  wss 3715  c0 4058   × cxp 5264  Rel wrel 5271
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-opab 4865  df-xp 5272  df-rel 5273
This theorem is referenced by: (None)
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