Theorem nrelv 5786

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 5270	. . 3 ⊢ ∅ ∈ V
2	1	notnoti 144	. 2 ⊢ ¬ ¬ ∅ ∈ V
3		0nelrel0 5721	. 2 ⊢ (Rel V → ¬ ∅ ∈ V)
4	2, 3	mto 200	1 ⊢ ¬ Rel V

Syntax hints:  ¬ wn 3   ∈ wcel 2143  Vcvv 3455  ∅c0 4286  Rel wrel 5666

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-opab 5174  df-xp 5667  df-rel 5668

This theorem is referenced by:  nfunv  6569  relintabex  44307