Theorem cnv0 6140

Description: The converse of the empty set. (Contributed by NM, 6-Apr-1998.) Remove dependency on ax-sep 5299, ax-nul 5306, ax-pr 5427. (Revised by KP, 25-Oct-2021.)

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 5197	. . . . . 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 5684	. . . . 5 ⊢ ◡∅ = {⟨𝑧, 𝑦⟩ ∣ 𝑦∅𝑧}
6		df-opab 5211	. . . . 5 ⊢ {⟨𝑧, 𝑦⟩ ∣ 𝑦∅𝑧} = {𝑥 ∣ ∃𝑧∃𝑦(𝑥 = ⟨𝑧, 𝑦⟩ ∧ 𝑦∅𝑧)}
7	5, 6	eqtri 2759	. . . 4 ⊢ ◡∅ = {𝑥 ∣ ∃𝑧∃𝑦(𝑥 = ⟨𝑧, 𝑦⟩ ∧ 𝑦∅𝑧)}
8	7	eqabri 2876	. . 3 ⊢ (𝑥 ∈ ◡∅ ↔ ∃𝑧∃𝑦(𝑥 = ⟨𝑧, 𝑦⟩ ∧ 𝑦∅𝑧))
9	4, 8	mtbir 323	. 2 ⊢ ¬ 𝑥 ∈ ◡∅
10	9	nel0 4350	1 ⊢ ◡∅ = ∅

Syntax hints:   ∧ wa 395   = wceq 1540  ∃wex 1780   ∈ wcel 2105  {cab 2708  ∅c0 4322  ⟨cop 4634   class class class wbr 5148  {copab 5210  ^◡ccnv 5675

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-12 2170  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-dif 3951  df-nul 4323  df-br 5149  df-opab 5211  df-cnv 5684

This theorem is referenced by:  xp0  6157  cnveq0  6196  co01  6260  funcnv0  6614  f1o00  6868  tpos0  8247  cnvfi  9186  oduleval  18252  ust0  24044  nghmfval  24559  isnghm  24560  1pthdlem1  29821  mptiffisupp  32348  tocycf  32712  tocyc01  32713  mthmval  35030  resnonrel  42806  cononrel1  42808  cononrel2  42809  cnvrcl0  42839  0cnf  45052