Theorem nrelv 5673

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 5211	. . 3 ⊢ ∅ ∈ V
2		0nelxp 5589	. . 3 ⊢ ¬ ∅ ∈ (V × V)
3		nelss 4030	. . 3 ⊢ ((∅ ∈ V ∧ ¬ ∅ ∈ (V × V)) → ¬ V ⊆ (V × V))
4	1, 2, 3	mp2an 690	. 2 ⊢ ¬ V ⊆ (V × V)
5		df-rel 5562	. 2 ⊢ (Rel V ↔ V ⊆ (V × V))
6	4, 5	mtbir 325	1 ⊢ ¬ Rel V

Syntax hints:  ¬ wn 3   ∈ wcel 2114  Vcvv 3494   ⊆ wss 3936  ∅c0 4291   × cxp 5553  Rel wrel 5560

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pr 5330

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-v 3496  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-opab 5129  df-xp 5561  df-rel 5562

This theorem is referenced by:  nfunv  6388  relintabex  39961