Theorem nrelv 5784

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 5282	. . 3 ⊢ ∅ ∈ V
2		0nelxp 5693	. . 3 ⊢ ¬ ∅ ∈ (V × V)
3		nelss 4029	. . 3 ⊢ ((∅ ∈ V ∧ ¬ ∅ ∈ (V × V)) → ¬ V ⊆ (V × V))
4	1, 2, 3	mp2an 692	. 2 ⊢ ¬ V ⊆ (V × V)
5		df-rel 5666	. 2 ⊢ (Rel V ↔ V ⊆ (V × V))
6	4, 5	mtbir 323	1 ⊢ ¬ Rel V

Syntax hints:  ¬ wn 3   ∈ wcel 2109  Vcvv 3464   ⊆ wss 3931  ∅c0 4313   × cxp 5657  Rel wrel 5664

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-ne 2934  df-v 3466  df-dif 3934  df-un 3936  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-opab 5187  df-xp 5665  df-rel 5666

This theorem is referenced by:  nfunv  6574  relintabex  43572