Theorem cnv0 5867

Description: The converse of the empty set. (Contributed by NM, 6-Apr-1998.) Remove dependency on ax-sep 5255, ax-nul 5267, ax-pr 5402. (Revised by KP, 25-Oct-2021.) Avoid ax-12 2215. (Revised by TM, 31-Jan-2026.)

Assertion

Ref	Expression
cnv0	⊢ ◡∅ = ∅

Proof of Theorem cnv0

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		br0 5158	. . . . . 6 ⊢ ¬ 𝑦∅𝑧
2	1	intnan 492	. . . . 5 ⊢ ¬ (𝑥 = ⟨𝑧, 𝑦⟩ ∧ 𝑦∅𝑧)
3	2	nex 1833	. . . 4 ⊢ ¬ ∃𝑦(𝑥 = ⟨𝑧, 𝑦⟩ ∧ 𝑦∅𝑧)
4	3	nex 1833	. . 3 ⊢ ¬ ∃𝑧∃𝑦(𝑥 = ⟨𝑧, 𝑦⟩ ∧ 𝑦∅𝑧)
5		df-cnv 5667	. . . . 5 ⊢ ◡∅ = {⟨𝑧, 𝑦⟩ ∣ 𝑦∅𝑧}
6	5	eleq2i 2854	. . . 4 ⊢ (𝑥 ∈ ◡∅ ↔ 𝑥 ∈ {⟨𝑧, 𝑦⟩ ∣ 𝑦∅𝑧})
7		elopabw 5508	. . . . 5 ⊢ (𝑥 ∈ V → (𝑥 ∈ {⟨𝑧, 𝑦⟩ ∣ 𝑦∅𝑧} ↔ ∃𝑧∃𝑦(𝑥 = ⟨𝑧, 𝑦⟩ ∧ 𝑦∅𝑧)))
8	7	elv 3458	. . . 4 ⊢ (𝑥 ∈ {⟨𝑧, 𝑦⟩ ∣ 𝑦∅𝑧} ↔ ∃𝑧∃𝑦(𝑥 = ⟨𝑧, 𝑦⟩ ∧ 𝑦∅𝑧))
9	6, 8	bitri 278	. . 3 ⊢ (𝑥 ∈ ◡∅ ↔ ∃𝑧∃𝑦(𝑥 = ⟨𝑧, 𝑦⟩ ∧ 𝑦∅𝑧))
10	4, 9	mtbir 326	. 2 ⊢ ¬ 𝑥 ∈ ◡∅
11	10	nel0 4305	1 ⊢ ◡∅ = ∅

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2145  Vcvv 3453  ∅c0 4282  ⟨cop 4593   class class class wbr 5107  {copab 5171  ^◡ccnv 5658

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 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-v 3455  df-dif 3905  df-nul 4283  df-br 5108  df-opab 5172  df-cnv 5667

This theorem is used by:  csbcnv  5870  xp0OLD  6154  cnveq0  6195  co01  6262  funcnv0  6603  f1o00  6857  tpos0  8257  cnvfi  9173  oduleval  18381  ust0  24447  nghmfval  24949  isnghm  24950  1pthdlem1  30591  mptiffisupp  33152  tocycf  33544  tocyc01  33545  vieta  34077  mthmval  36141  resnonrel  44419  cononrel1  44421  cononrel2  44422  cnvrcl0  44452  0cnf  46692