Theorem nrelv 5668

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 5204	. . 3 ⊢ ∅ ∈ V
2		0nelxp 5584	. . 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 5557	. 2 ⊢ (Rel V ↔ V ⊆ (V × V))
6	4, 5	mtbir 325	1 ⊢ ¬ Rel V

Syntax hints:  ¬ wn 3   ∈ wcel 2110  Vcvv 3495   ⊆ wss 3936  ∅c0 4291   × cxp 5548  Rel wrel 5555

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-sep 5196  ax-nul 5203  ax-pr 5322

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-v 3497  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4562  df-pr 4564  df-op 4568  df-opab 5122  df-xp 5556  df-rel 5557

This theorem is referenced by:  nfunv  6383  relintabex  39934