Theorem nrelv 5471

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 5026	. . 3 ⊢ ∅ ∈ V
2		0nelxp 5389	. . 3 ⊢ ¬ ∅ ∈ (V × V)
3		nelss 3882	. . 3 ⊢ ((∅ ∈ V ∧ ¬ ∅ ∈ (V × V)) → ¬ V ⊆ (V × V))
4	1, 2, 3	mp2an 682	. 2 ⊢ ¬ V ⊆ (V × V)
5		df-rel 5362	. 2 ⊢ (Rel V ↔ V ⊆ (V × V))
6	4, 5	mtbir 315	1 ⊢ ¬ Rel V

Syntax hints:  ¬ wn 3   ∈ wcel 2106  Vcvv 3397   ⊆ wss 3791  ∅c0 4140   × cxp 5353  Rel wrel 5360

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-ext 2753  ax-sep 5017  ax-nul 5025  ax-pr 5138

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ne 2969  df-v 3399  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-opab 4949  df-xp 5361  df-rel 5362

This theorem is referenced by:  nfunv  6168