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Theorem cnv0 6163
Description: The converse of the empty set. (Contributed by NM, 6-Apr-1998.) Remove dependency on ax-sep 5302, ax-nul 5312, ax-pr 5438. (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 br0 5197 . . . . . 6 ¬ 𝑦𝑧
21intnan 486 . . . . 5 ¬ (𝑥 = ⟨𝑧, 𝑦⟩ ∧ 𝑦𝑧)
32nex 1797 . . . 4 ¬ ∃𝑦(𝑥 = ⟨𝑧, 𝑦⟩ ∧ 𝑦𝑧)
43nex 1797 . . 3 ¬ ∃𝑧𝑦(𝑥 = ⟨𝑧, 𝑦⟩ ∧ 𝑦𝑧)
5 df-cnv 5697 . . . . 5 ∅ = {⟨𝑧, 𝑦⟩ ∣ 𝑦𝑧}
6 df-opab 5211 . . . . 5 {⟨𝑧, 𝑦⟩ ∣ 𝑦𝑧} = {𝑥 ∣ ∃𝑧𝑦(𝑥 = ⟨𝑧, 𝑦⟩ ∧ 𝑦𝑧)}
75, 6eqtri 2763 . . . 4 ∅ = {𝑥 ∣ ∃𝑧𝑦(𝑥 = ⟨𝑧, 𝑦⟩ ∧ 𝑦𝑧)}
87eqabri 2883 . . 3 (𝑥∅ ↔ ∃𝑧𝑦(𝑥 = ⟨𝑧, 𝑦⟩ ∧ 𝑦𝑧))
94, 8mtbir 323 . 2 ¬ 𝑥
109nel0 4360 1 ∅ = ∅
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1537  wex 1776  wcel 2106  {cab 2712  c0 4339  cop 4637   class class class wbr 5148  {copab 5210  ccnv 5688
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-dif 3966  df-nul 4340  df-br 5149  df-opab 5211  df-cnv 5697
This theorem is referenced by:  xp0  6180  cnveq0  6219  co01  6283  funcnv0  6634  f1o00  6884  tpos0  8280  cnvfi  9215  oduleval  18346  ust0  24244  nghmfval  24759  isnghm  24760  1pthdlem1  30164  mptiffisupp  32708  tocycf  33120  tocyc01  33121  mthmval  35560  resnonrel  43582  cononrel1  43584  cononrel2  43585  cnvrcl0  43615  0cnf  45833
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