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Theorem cnv0 5570
Description: The converse of the empty set. (Contributed by NM, 6-Apr-1998.) Remove dependency on ax-sep 4814, ax-nul 4822, ax-pr 4936. (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 4734 . . . . . 6 ¬ 𝑦𝑧
21intnan 980 . . . . 5 ¬ (𝑥 = ⟨𝑧, 𝑦⟩ ∧ 𝑦𝑧)
32nex 1771 . . . 4 ¬ ∃𝑦(𝑥 = ⟨𝑧, 𝑦⟩ ∧ 𝑦𝑧)
43nex 1771 . . 3 ¬ ∃𝑧𝑦(𝑥 = ⟨𝑧, 𝑦⟩ ∧ 𝑦𝑧)
5 df-cnv 5151 . . . . 5 ∅ = {⟨𝑧, 𝑦⟩ ∣ 𝑦𝑧}
6 df-opab 4746 . . . . 5 {⟨𝑧, 𝑦⟩ ∣ 𝑦𝑧} = {𝑥 ∣ ∃𝑧𝑦(𝑥 = ⟨𝑧, 𝑦⟩ ∧ 𝑦𝑧)}
75, 6eqtri 2673 . . . 4 ∅ = {𝑥 ∣ ∃𝑧𝑦(𝑥 = ⟨𝑧, 𝑦⟩ ∧ 𝑦𝑧)}
87abeq2i 2764 . . 3 (𝑥∅ ↔ ∃𝑧𝑦(𝑥 = ⟨𝑧, 𝑦⟩ ∧ 𝑦𝑧))
94, 8mtbir 312 . 2 ¬ 𝑥
109nel0 3965 1 ∅ = ∅
Colors of variables: wff setvar class
Syntax hints:  wa 383   = wceq 1523  wex 1744  wcel 2030  {cab 2637  c0 3948  cop 4216   class class class wbr 4685  {copab 4745  ccnv 5142
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-v 3233  df-dif 3610  df-nul 3949  df-br 4686  df-opab 4746  df-cnv 5151
This theorem is referenced by:  xp0  5587  cnveq0  5626  co01  5688  funcnv0  5993  f10  6207  f1o00  6209  tpos0  7427  oduleval  17178  ust0  22070  nghmfval  22573  isnghm  22574  1pthdlem1  27113  mthmval  31598  resnonrel  38215  cononrel1  38217  cononrel2  38218  cnvrcl0  38249  0cnf  40408  mbf0  40491
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