Theorem nrelv 5806

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 5311	. . 3 ⊢ ∅ ∈ V
2		0nelxp 5716	. . 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 5689	. 2 ⊢ (Rel V ↔ V ⊆ (V × V))
6	4, 5	mtbir 322	1 ⊢ ¬ Rel V

Syntax hints:  ¬ wn 3   ∈ wcel 2098  Vcvv 3473   ⊆ wss 3949  ∅c0 4326   × cxp 5680  Rel wrel 5687

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-opab 5215  df-xp 5688  df-rel 5689

This theorem is referenced by:  nfunv  6591  relintabex  43042