Theorem nrelv 5791

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

Step	Hyp	Ref	Expression
1		0ex 5298	. . 3 ⊢ ∅ ∈ V
2		0nelxp 5701	. . 3 ⊢ ¬ ∅ ∈ (V × V)
3		nelss 4040	. . 3 ⊢ ((∅ ∈ V ∧ ¬ ∅ ∈ (V × V)) → ¬ V ⊆ (V × V))
4	1, 2, 3	mp2an 689	. 2 ⊢ ¬ V ⊆ (V × V)
5		df-rel 5674	. 2 ⊢ (Rel V ↔ V ⊆ (V × V))
6	4, 5	mtbir 323	1 ⊢ ¬ Rel V

Syntax hints:  ¬ wn 3   ∈ wcel 2098  Vcvv 3466   ⊆ wss 3941  ∅c0 4315   × cxp 5665  Rel wrel 5672

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ne 2933  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-opab 5202  df-xp 5673  df-rel 5674

This theorem is referenced by:  nfunv  6572  relintabex  42846