Theorem nrelvOLD 5785

Description: Obsolete version of nrelv 5784 as of 10-Jun-2026. (Contributed by Thierry Arnoux, 23-Jan-2022.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
nrelvOLD	⊢ ¬ Rel V

Proof of Theorem nrelvOLD

Step	Hyp	Ref	Expression
1		0ex 5268	. . 3 ⊢ ∅ ∈ V
2		0nelxp 5693	. . 3 ⊢ ¬ ∅ ∈ (V × V)
3		nelss 4000	. . 3 ⊢ ((∅ ∈ V ∧ ¬ ∅ ∈ (V × V)) → ¬ V ⊆ (V × V))
4	1, 2, 3	mp2an 705	. 2 ⊢ ¬ V ⊆ (V × V)
5		df-rel 5666	. 2 ⊢ (Rel V ↔ V ⊆ (V × V))
6	4, 5	mtbir 326	1 ⊢ ¬ Rel V

This proof depends on syntax axioms:  ¬ wn 3   ∈ wcel 2145  Vcvv 3453   ⊆ wss 3902  ∅c0 4282   × cxp 5657  Rel wrel 5664

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-opab 5172  df-xp 5665  df-rel 5666

This theorem is used by: (None)