Theorem nrelv 5781

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 5263	. . 3 ⊢ ∅ ∈ V
2	1	notnoti 144	. 2 ⊢ ¬ ¬ ∅ ∈ V
3		0nelrel0 5714	. 2 ⊢ (Rel V → ¬ ∅ ∈ V)
4	2, 3	mto 200	1 ⊢ ¬ Rel V

This proof depends on syntax axioms:  ¬ wn 3   ∈ wcel 2146  Vcvv 3453  ∅c0 4280  Rel wrel 5659

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-ext 2735  ax-sep 5251  ax-nul 5262  ax-pr 5394

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-rab 3416  df-v 3455  df-ss 3900  df-dif 4049  df-in 4093  df-un 4153  df-nul 4281  df-if 4485  df-sn 4588  df-pr 4590  df-op 4838  df-opab 5170  df-xp 5660  df-rel 5661

This theorem is used by:  nfunv  6581  relintabex  44722