Theorem nrelv 5800

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 5307	. . 3 ⊢ ∅ ∈ V
2		0nelxp 5710	. . 3 ⊢ ¬ ∅ ∈ (V × V)
3		nelss 4047	. . 3 ⊢ ((∅ ∈ V ∧ ¬ ∅ ∈ (V × V)) → ¬ V ⊆ (V × V))
4	1, 2, 3	mp2an 690	. 2 ⊢ ¬ V ⊆ (V × V)
5		df-rel 5683	. 2 ⊢ (Rel V ↔ V ⊆ (V × V))
6	4, 5	mtbir 322	1 ⊢ ¬ Rel V

Syntax hints:  ¬ wn 3   ∈ wcel 2106  Vcvv 3474   ⊆ wss 3948  ∅c0 4322   × cxp 5674  Rel wrel 5681

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-opab 5211  df-xp 5682  df-rel 5683

This theorem is referenced by:  nfunv  6581  relintabex  42322