Theorem nrelvOLD 5774

Description: Obsolete version of nrelv 5773 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 5258	. . 3 ⊢ ∅ ∈ V
2		0nelxp 5682	. . 3 ⊢ ¬ ∅ ∈ (V × V)
3		nelss 4003	. . 3 ⊢ ((∅ ∈ V ∧ ¬ ∅ ∈ (V × V)) → ¬ V ⊆ (V × V))
4	1, 2, 3	mp2an 702	. 2 ⊢ ¬ V ⊆ (V × V)
5		df-rel 5655	. 2 ⊢ (Rel V ↔ V ⊆ (V × V))
6	4, 5	mtbir 325	1 ⊢ ¬ Rel V

Syntax hints:  ¬ wn 3   ∈ wcel 2143  Vcvv 3455   ⊆ wss 3905  ∅c0 4286   × cxp 5646  Rel wrel 5653

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5247  ax-nul 5257  ax-pr 5391

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-sb 2092  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-rab 3416  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-opab 5164  df-xp 5654  df-rel 5655

This theorem is referenced by: (None)