Theorem nrelv 5244

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 4790	. . 3 ⊢ ∅ ∈ V
2		0nelxp 5143	. . 3 ⊢ ¬ ∅ ∈ (V × V)
3		nelss 3664	. . 3 ⊢ ((∅ ∈ V ∧ ¬ ∅ ∈ (V × V)) → ¬ V ⊆ (V × V))
4	1, 2, 3	mp2an 708	. 2 ⊢ ¬ V ⊆ (V × V)
5		df-rel 5121	. 2 ⊢ (Rel V ↔ V ⊆ (V × V))
6	4, 5	mtbir 313	1 ⊢ ¬ Rel V

Syntax hints:  ¬ wn 3   ∈ wcel 1990  Vcvv 3200   ⊆ wss 3574  ∅c0 3915   × cxp 5112  Rel wrel 5119

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-opab 4713  df-xp 5120  df-rel 5121

This theorem is referenced by: (None)