Theorem nrelv 5780

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 5264	. . 3 ⊢ ∅ ∈ V
2	1	notnoti 144	. 2 ⊢ ¬ ¬ ∅ ∈ V
3		0nelrel0 5715	. 2 ⊢ (Rel V → ¬ ∅ ∈ V)
4	2, 3	mto 200	1 ⊢ ¬ Rel V

This proof depends on syntax axioms:  ¬ wn 3   ∈ wcel 2145  Vcvv 3450  ∅c0 4279  Rel wrel 5660

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-opab 5168  df-xp 5661  df-rel 5662

This theorem is used by:  nfunv  6567  relintabex  44422