Theorem nrelv 5772

Description: The universal class is not a relation. (Contributed by Thierry Arnoux, 23-Jan-2022.) (Proof shortened by Umit Teoman Dogan, 10-Jun-2026.)

Assertion

Ref	Expression
nrelv	⊢ ¬ Rel V

Proof of Theorem nrelv

Step	Hyp	Ref	Expression
1		0ex 5257	. . 3 ⊢ ∅ ∈ V
2	1	notnoti 143	. 2 ⊢ ¬ ¬ ∅ ∈ V
3		0nelrel0 5707	. 2 ⊢ (Rel V → ¬ ∅ ∈ V)
4	2, 3	mto 199	1 ⊢ ¬ Rel V

Syntax hints:  ¬ wn 3   ∈ wcel 2142  Vcvv 3454  ∅c0 4285  Rel wrel 5652

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5246  ax-nul 5256  ax-pr 5390

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-opab 5163  df-xp 5653  df-rel 5654

This theorem is referenced by:  nfunv  6554  relintabex  44157