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 5260	. . 3 ⊢ ∅ ∈ V
2		0nelxp 5681	. . 3 ⊢ ¬ ∅ ∈ (V × V)
3		nelss 3996	. . 3 ⊢ ((∅ ∈ V ∧ ¬ ∅ ∈ (V × V)) → ¬ V ⊆ (V × V))
4	1, 2, 3	mp2an 705	. 2 ⊢ ¬ V ⊆ (V × V)
5		df-rel 5654	. 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 3450   ⊆ wss 3898  ∅c0 4278   × cxp 5645  Rel wrel 5652

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 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390

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 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-rab 3413  df-v 3452  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-opab 5167  df-xp 5653  df-rel 5654

This theorem is used by: (None)