Theorem cnv0 6134

Description: The converse of the empty set. (Contributed by NM, 6-Apr-1998.) Remove dependency on ax-sep 5271, ax-nul 5281, ax-pr 5407. (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 5173	. . . . . 6 ⊢ ¬ 𝑦∅𝑧
2	1	intnan 486	. . . . 5 ⊢ ¬ (𝑥 = ⟨𝑧, 𝑦⟩ ∧ 𝑦∅𝑧)
3	2	nex 1800	. . . 4 ⊢ ¬ ∃𝑦(𝑥 = ⟨𝑧, 𝑦⟩ ∧ 𝑦∅𝑧)
4	3	nex 1800	. . 3 ⊢ ¬ ∃𝑧∃𝑦(𝑥 = ⟨𝑧, 𝑦⟩ ∧ 𝑦∅𝑧)
5		df-cnv 5667	. . . . 5 ⊢ ◡∅ = {⟨𝑧, 𝑦⟩ ∣ 𝑦∅𝑧}
6		df-opab 5187	. . . . 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 2879	. . 3 ⊢ (𝑥 ∈ ◡∅ ↔ ∃𝑧∃𝑦(𝑥 = ⟨𝑧, 𝑦⟩ ∧ 𝑦∅𝑧))
9	4, 8	mtbir 323	. 2 ⊢ ¬ 𝑥 ∈ ◡∅
10	9	nel0 4334	1 ⊢ ◡∅ = ∅

Syntax hints:   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2109  {cab 2714  ∅c0 4313  ⟨cop 4612   class class class wbr 5124  {copab 5186  ^◡ccnv 5658

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-12 2178  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-dif 3934  df-nul 4314  df-br 5125  df-opab 5187  df-cnv 5667

This theorem is referenced by:  xp0  6152  cnveq0  6191  co01  6255  funcnv0  6607  f1o00  6858  tpos0  8260  cnvfi  9195  oduleval  18306  ust0  24163  nghmfval  24666  isnghm  24667  1pthdlem1  30121  mptiffisupp  32675  tocycf  33133  tocyc01  33134  mthmval  35602  resnonrel  43583  cononrel1  43585  cononrel2  43586  cnvrcl0  43616  0cnf  45873