Theorem cnv0 6086

Description: The converse of the empty set. (Contributed by NM, 6-Apr-1998.) Remove dependency on ax-sep 5232, ax-nul 5242, ax-pr 5368. (Revised by KP, 25-Oct-2021.) Avoid ax-12 2180. (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 5138	. . . . . 6 ⊢ ¬ 𝑦∅𝑧
2	1	intnan 486	. . . . 5 ⊢ ¬ (𝑥 = ⟨𝑧, 𝑦⟩ ∧ 𝑦∅𝑧)
3	2	nex 1801	. . . 4 ⊢ ¬ ∃𝑦(𝑥 = ⟨𝑧, 𝑦⟩ ∧ 𝑦∅𝑧)
4	3	nex 1801	. . 3 ⊢ ¬ ∃𝑧∃𝑦(𝑥 = ⟨𝑧, 𝑦⟩ ∧ 𝑦∅𝑧)
5		df-cnv 5622	. . . . 5 ⊢ ◡∅ = {⟨𝑧, 𝑦⟩ ∣ 𝑦∅𝑧}
6	5	eleq2i 2823	. . . 4 ⊢ (𝑥 ∈ ◡∅ ↔ 𝑥 ∈ {⟨𝑧, 𝑦⟩ ∣ 𝑦∅𝑧})
7		elopabw 5464	. . . . 5 ⊢ (𝑥 ∈ V → (𝑥 ∈ {⟨𝑧, 𝑦⟩ ∣ 𝑦∅𝑧} ↔ ∃𝑧∃𝑦(𝑥 = ⟨𝑧, 𝑦⟩ ∧ 𝑦∅𝑧)))
8	7	elv 3441	. . . 4 ⊢ (𝑥 ∈ {⟨𝑧, 𝑦⟩ ∣ 𝑦∅𝑧} ↔ ∃𝑧∃𝑦(𝑥 = ⟨𝑧, 𝑦⟩ ∧ 𝑦∅𝑧))
9	6, 8	bitri 275	. . 3 ⊢ (𝑥 ∈ ◡∅ ↔ ∃𝑧∃𝑦(𝑥 = ⟨𝑧, 𝑦⟩ ∧ 𝑦∅𝑧))
10	4, 9	mtbir 323	. 2 ⊢ ¬ 𝑥 ∈ ◡∅
11	10	nel0 4301	1 ⊢ ◡∅ = ∅

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1541  ∃wex 1780   ∈ wcel 2111  Vcvv 3436  ∅c0 4280  ⟨cop 4579   class class class wbr 5089  {copab 5151  ^◡ccnv 5613

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-v 3438  df-dif 3900  df-nul 4281  df-br 5090  df-opab 5152  df-cnv 5622

This theorem is referenced by:  xp0OLD  6105  cnveq0  6144  co01  6209  funcnv0  6547  f1o00  6798  tpos0  8186  cnvfi  9085  oduleval  18195  ust0  24135  nghmfval  24637  isnghm  24638  1pthdlem1  30115  mptiffisupp  32674  tocycf  33086  tocyc01  33087  mthmval  35619  resnonrel  43684  cononrel1  43686  cononrel2  43687  cnvrcl0  43717  0cnf  45974