Theorem nrelv 5637

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 5175	. . 3 ⊢ ∅ ∈ V
2		0nelxp 5553	. . 3 ⊢ ¬ ∅ ∈ (V × V)
3		nelss 3978	. . 3 ⊢ ((∅ ∈ V ∧ ¬ ∅ ∈ (V × V)) → ¬ V ⊆ (V × V))
4	1, 2, 3	mp2an 691	. 2 ⊢ ¬ V ⊆ (V × V)
5		df-rel 5526	. 2 ⊢ (Rel V ↔ V ⊆ (V × V))
6	4, 5	mtbir 326	1 ⊢ ¬ Rel V

Syntax hints:  ¬ wn 3   ∈ wcel 2111  Vcvv 3441   ⊆ wss 3881  ∅c0 4243   × cxp 5517  Rel wrel 5524

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-opab 5093  df-xp 5525  df-rel 5526

This theorem is referenced by:  nfunv  6357  relintabex  40281