Theorem nrelv 5735

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 5240	. . 3 ⊢ ∅ ∈ V
2		0nelxp 5645	. . 3 ⊢ ¬ ∅ ∈ (V × V)
3		nelss 3995	. . 3 ⊢ ((∅ ∈ V ∧ ¬ ∅ ∈ (V × V)) → ¬ V ⊆ (V × V))
4	1, 2, 3	mp2an 692	. 2 ⊢ ¬ V ⊆ (V × V)
5		df-rel 5618	. 2 ⊢ (Rel V ↔ V ⊆ (V × V))
6	4, 5	mtbir 323	1 ⊢ ¬ Rel V

Syntax hints:  ¬ wn 3   ∈ wcel 2111  Vcvv 3436   ⊆ wss 3897  ∅c0 4278   × cxp 5609  Rel wrel 5616

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 2113  ax-9 2121  ax-ext 2703  ax-sep 5229  ax-nul 5239  ax-pr 5365

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 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4279  df-if 4471  df-sn 4572  df-pr 4574  df-op 4578  df-opab 5149  df-xp 5617  df-rel 5618

This theorem is referenced by:  nfunv  6509  relintabex  43614