Theorem nrelvOLD 5787

Description: Obsolete version of nrelv 5786 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 5269	. . 3 ⊢ ∅ ∈ V
2		0nelxp 5695	. . 3 ⊢ ¬ ∅ ∈ (V × V)
3		nelss 4002	. . 3 ⊢ ((∅ ∈ V ∧ ¬ ∅ ∈ (V × V)) → ¬ V ⊆ (V × V))
4	1, 2, 3	mp2an 704	. 2 ⊢ ¬ V ⊆ (V × V)
5		df-rel 5668	. 2 ⊢ (Rel V ↔ V ⊆ (V × V))
6	4, 5	mtbir 326	1 ⊢ ¬ Rel V

Syntax hints:  ¬ wn 3   ∈ wcel 2141  Vcvv 3453   ⊆ wss 3904  ∅c0 4285   × cxp 5659  Rel wrel 5666

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5256  ax-nul 5268  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-sb 2095  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-rab 3415  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-opab 5173  df-xp 5667  df-rel 5668

This theorem is referenced by: (None)