Theorem nrelvOLD 5762

Description: Obsolete version of nrelv 5761 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 5247	. . 3 ⊢ ∅ ∈ V
2		0nelxp 5670	. . 3 ⊢ ¬ ∅ ∈ (V × V)
3		nelss 3993	. . 3 ⊢ ((∅ ∈ V ∧ ¬ ∅ ∈ (V × V)) → ¬ V ⊆ (V × V))
4	1, 2, 3	mp2an 700	. 2 ⊢ ¬ V ⊆ (V × V)
5		df-rel 5643	. 2 ⊢ (Rel V ↔ V ⊆ (V × V))
6	4, 5	mtbir 325	1 ⊢ ¬ Rel V

Syntax hints:  ¬ wn 3   ∈ wcel 2132  Vcvv 3444   ⊆ wss 3895  ∅c0 4276   × cxp 5634  Rel wrel 5641

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1805  ax-4 1819  ax-5 1920  ax-6 1977  ax-7 2018  ax-8 2134  ax-9 2142  ax-ext 2724  ax-sep 5236  ax-nul 5246  ax-pr 5380

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3an 1097  df-tru 1553  df-fal 1563  df-ex 1790  df-sb 2081  df-clab 2731  df-cleq 2744  df-clel 2827  df-ne 2948  df-rab 3405  df-v 3446  df-dif 3898  df-un 3900  df-in 3902  df-ss 3912  df-nul 4277  df-if 4471  df-sn 4573  df-pr 4575  df-op 4579  df-opab 5153  df-xp 5642  df-rel 5643

This theorem is referenced by: (None)