Theorem nrelv 5743

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 5229	. . 3 ⊢ ∅ ∈ V
2		0nelxp 5652	. . 3 ⊢ ¬ ∅ ∈ (V × V)
3		nelss 3980	. . 3 ⊢ ((∅ ∈ V ∧ ¬ ∅ ∈ (V × V)) → ¬ V ⊆ (V × V))
4	1, 2, 3	mp2an 698	. 2 ⊢ ¬ V ⊆ (V × V)
5		df-rel 5625	. 2 ⊢ (Rel V ↔ V ⊆ (V × V))
6	4, 5	mtbir 324	1 ⊢ ¬ Rel V

Syntax hints:  ¬ wn 3   ∈ wcel 2119  Vcvv 3431   ⊆ wss 3883  ∅c0 4261   × cxp 5616  Rel wrel 5623

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ne 2935  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-opab 5135  df-xp 5624  df-rel 5625

This theorem is referenced by:  nfunv  6518  relintabex  44025