Theorem nrelv 5749

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 5242	. . 3 ⊢ ∅ ∈ V
2		0nelxp 5658	. . 3 ⊢ ¬ ∅ ∈ (V × V)
3		nelss 3988	. . 3 ⊢ ((∅ ∈ V ∧ ¬ ∅ ∈ (V × V)) → ¬ V ⊆ (V × V))
4	1, 2, 3	mp2an 693	. 2 ⊢ ¬ V ⊆ (V × V)
5		df-rel 5631	. 2 ⊢ (Rel V ↔ V ⊆ (V × V))
6	4, 5	mtbir 323	1 ⊢ ¬ Rel V

Syntax hints:  ¬ wn 3   ∈ wcel 2114  Vcvv 3430   ⊆ wss 3890  ∅c0 4274   × cxp 5622  Rel wrel 5629

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-opab 5149  df-xp 5630  df-rel 5631

This theorem is referenced by:  nfunv  6525  relintabex  44026