Theorem cnv0 6172

Description: The converse of the empty set. (Contributed by NM, 6-Apr-1998.) Remove dependency on ax-sep 5317, ax-nul 5324, ax-pr 5447. (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 5215	. . . . . 6 ⊢ ¬ 𝑦∅𝑧
2	1	intnan 486	. . . . 5 ⊢ ¬ (𝑥 = ⟨𝑧, 𝑦⟩ ∧ 𝑦∅𝑧)
3	2	nex 1798	. . . 4 ⊢ ¬ ∃𝑦(𝑥 = ⟨𝑧, 𝑦⟩ ∧ 𝑦∅𝑧)
4	3	nex 1798	. . 3 ⊢ ¬ ∃𝑧∃𝑦(𝑥 = ⟨𝑧, 𝑦⟩ ∧ 𝑦∅𝑧)
5		df-cnv 5708	. . . . 5 ⊢ ◡∅ = {⟨𝑧, 𝑦⟩ ∣ 𝑦∅𝑧}
6		df-opab 5229	. . . . 5 ⊢ {⟨𝑧, 𝑦⟩ ∣ 𝑦∅𝑧} = {𝑥 ∣ ∃𝑧∃𝑦(𝑥 = ⟨𝑧, 𝑦⟩ ∧ 𝑦∅𝑧)}
7	5, 6	eqtri 2768	. . . 4 ⊢ ◡∅ = {𝑥 ∣ ∃𝑧∃𝑦(𝑥 = ⟨𝑧, 𝑦⟩ ∧ 𝑦∅𝑧)}
8	7	eqabri 2888	. . 3 ⊢ (𝑥 ∈ ◡∅ ↔ ∃𝑧∃𝑦(𝑥 = ⟨𝑧, 𝑦⟩ ∧ 𝑦∅𝑧))
9	4, 8	mtbir 323	. 2 ⊢ ¬ 𝑥 ∈ ◡∅
10	9	nel0 4377	1 ⊢ ◡∅ = ∅

Syntax hints:   ∧ wa 395   = wceq 1537  ∃wex 1777   ∈ wcel 2108  {cab 2717  ∅c0 4352  ⟨cop 4654   class class class wbr 5166  {copab 5228  ^◡ccnv 5699

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-12 2178  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-dif 3979  df-nul 4353  df-br 5167  df-opab 5229  df-cnv 5708

This theorem is referenced by:  xp0  6189  cnveq0  6228  co01  6292  funcnv0  6644  f1o00  6897  tpos0  8297  cnvfi  9243  oduleval  18359  ust0  24249  nghmfval  24764  isnghm  24765  1pthdlem1  30167  mptiffisupp  32705  tocycf  33110  tocyc01  33111  mthmval  35543  resnonrel  43554  cononrel1  43556  cononrel2  43557  cnvrcl0  43587  0cnf  45798