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Theorem cnv0 5454
 Description: The converse of the empty set. (Contributed by NM, 6-Apr-1998.) Remove dependency on ax-sep 4709, ax-nul 4717, ax-pr 4833. (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.
StepHypRef Expression
1 eq0 3888 . 2 (∅ = ∅ ↔ ∀𝑥 ¬ 𝑥∅)
2 br0 4631 . . . . . 6 ¬ 𝑦𝑧
32intnan 951 . . . . 5 ¬ (𝑥 = ⟨𝑧, 𝑦⟩ ∧ 𝑦𝑧)
43nex 1722 . . . 4 ¬ ∃𝑦(𝑥 = ⟨𝑧, 𝑦⟩ ∧ 𝑦𝑧)
54nex 1722 . . 3 ¬ ∃𝑧𝑦(𝑥 = ⟨𝑧, 𝑦⟩ ∧ 𝑦𝑧)
6 df-cnv 5046 . . . . 5 ∅ = {⟨𝑧, 𝑦⟩ ∣ 𝑦𝑧}
7 df-opab 4644 . . . . 5 {⟨𝑧, 𝑦⟩ ∣ 𝑦𝑧} = {𝑥 ∣ ∃𝑧𝑦(𝑥 = ⟨𝑧, 𝑦⟩ ∧ 𝑦𝑧)}
86, 7eqtri 2632 . . . 4 ∅ = {𝑥 ∣ ∃𝑧𝑦(𝑥 = ⟨𝑧, 𝑦⟩ ∧ 𝑦𝑧)}
98abeq2i 2722 . . 3 (𝑥∅ ↔ ∃𝑧𝑦(𝑥 = ⟨𝑧, 𝑦⟩ ∧ 𝑦𝑧))
105, 9mtbir 312 . 2 ¬ 𝑥
111, 10mpgbir 1717 1 ∅ = ∅
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  {cab 2596  ∅c0 3874  ⟨cop 4131   class class class wbr 4583  {copab 4642  ◡ccnv 5037 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-dif 3543  df-nul 3875  df-br 4584  df-opab 4644  df-cnv 5046 This theorem is referenced by:  xp0  5471  cnveq0  5509  co01  5567  funcnv0  5869  f10  6081  f1o00  6083  tpos0  7269  oduleval  16954  ust0  21833  nghmfval  22336  isnghm  22337  1pthonlem1  26119  mthmval  30726  resnonrel  36917  cononrel1  36919  cononrel2  36920  cnvrcl0  36951  0cnf  38762  mbf0  38849  1pthdlem1  41302
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