Theorem nrelv 5746

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 5249	. . 3 ⊢ ∅ ∈ V
2		0nelxp 5655	. . 3 ⊢ ¬ ∅ ∈ (V × V)
3		nelss 3996	. . 3 ⊢ ((∅ ∈ V ∧ ¬ ∅ ∈ (V × V)) → ¬ V ⊆ (V × V))
4	1, 2, 3	mp2an 692	. 2 ⊢ ¬ V ⊆ (V × V)
5		df-rel 5628	. 2 ⊢ (Rel V ↔ V ⊆ (V × V))
6	4, 5	mtbir 323	1 ⊢ ¬ Rel V

Syntax hints:  ¬ wn 3   ∈ wcel 2113  Vcvv 3437   ⊆ wss 3898  ∅c0 4282   × cxp 5619  Rel wrel 5626

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-ne 2930  df-v 3439  df-dif 3901  df-un 3903  df-ss 3915  df-nul 4283  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-opab 5158  df-xp 5627  df-rel 5628

This theorem is referenced by:  nfunv  6522  relintabex  43738