Theorem nrelvOLD 5783

Description: Obsolete version of nrelv 5782 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 5264	. . 3 ⊢ ∅ ∈ V
2		0nelxp 5689	. . 3 ⊢ ¬ ∅ ∈ (V × V)
3		nelss 3979	. . 3 ⊢ ((∅ ∈ V ∧ ¬ ∅ ∈ (V × V)) → ¬ V ⊆ (V × V))
4	1, 2, 3	mp2an 705	. 2 ⊢ ¬ V ⊆ (V × V)
5		df-rel 5662	. 2 ⊢ (Rel V ↔ V ⊆ (V × V))
6	4, 5	mtbir 326	1 ⊢ ¬ Rel V

This proof depends on syntax axioms:  ¬ wn 3   ∈ wcel 2147  Vcvv 3454   ⊆ wss 3900  ∅c0 4281   × cxp 5653  Rel wrel 5660

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pr 5395

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 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-opab 5171  df-xp 5661  df-rel 5662

This theorem is used by: (None)