Theorem cnv0 5860

Description: The converse of the empty set. (Contributed by NM, 6-Apr-1998.) Remove dependency on ax-sep 5248, ax-nul 5259, ax-pr 5390. (Revised by KP, 25-Oct-2021.) Avoid ax-12 2212. (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 5153	. . . . . 6 ⊢ ¬ 𝑦∅𝑧
2	1	intnan 492	. . . . 5 ⊢ ¬ (𝑥 = ⟨𝑧, 𝑦⟩ ∧ 𝑦∅𝑧)
3	2	nex 1832	. . . 4 ⊢ ¬ ∃𝑦(𝑥 = ⟨𝑧, 𝑦⟩ ∧ 𝑦∅𝑧)
4	3	nex 1832	. . 3 ⊢ ¬ ∃𝑧∃𝑦(𝑥 = ⟨𝑧, 𝑦⟩ ∧ 𝑦∅𝑧)
5		df-cnv 5658	. . . . 5 ⊢ ◡∅ = {⟨𝑧, 𝑦⟩ ∣ 𝑦∅𝑧}
6	5	eleq2i 2852	. . . 4 ⊢ (𝑥 ∈ ◡∅ ↔ 𝑥 ∈ {⟨𝑧, 𝑦⟩ ∣ 𝑦∅𝑧})
7		elopabw 5499	. . . . 5 ⊢ (𝑥 ∈ V → (𝑥 ∈ {⟨𝑧, 𝑦⟩ ∣ 𝑦∅𝑧} ↔ ∃𝑧∃𝑦(𝑥 = ⟨𝑧, 𝑦⟩ ∧ 𝑦∅𝑧)))
8	7	elv 3455	. . . 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 4301	1 ⊢ ◡∅ = ∅

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1569  ∃wex 1811   ∈ wcel 2144  Vcvv 3450  ∅c0 4278  ⟨cop 4589   class class class wbr 5102  {copab 5166  ^◡ccnv 5649

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-dif 4047  df-nul 4279  df-br 5103  df-opab 5167  df-cnv 5658

This theorem is used by:  csbcnv  5863  xp0OLD  6148  cnveq0  6190  co01  6263  funcnv0  6606  f1o00  6860  tpos0  8273  cnvfi  9191  oduleval  18463  ust0  24539  nghmfval  25041  isnghm  25042  1pthdlem1  30726  mptiffisupp  33286  tocycf  33678  tocyc01  33679  vieta  34212  mthmval  36340  resnonrel  44592  cononrel1  44594  cononrel2  44595  cnvrcl0  44625  0cnf  46887