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Theorem dm0 4638
Description: The domain of the empty set is empty. Part of Theorem 3.8(v) of [Monk1] p. 36. (Contributed by NM, 4-Jul-1994.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
dm0 dom ∅ = ∅

Proof of Theorem dm0
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eq0 3299 . 2 (dom ∅ = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ dom ∅)
2 noel 3288 . . . 4 ¬ ⟨𝑥, 𝑦⟩ ∈ ∅
32nex 1434 . . 3 ¬ ∃𝑦𝑥, 𝑦⟩ ∈ ∅
4 vex 2622 . . . 4 𝑥 ∈ V
54eldm2 4622 . . 3 (𝑥 ∈ dom ∅ ↔ ∃𝑦𝑥, 𝑦⟩ ∈ ∅)
63, 5mtbir 631 . 2 ¬ 𝑥 ∈ dom ∅
71, 6mpgbir 1387 1 dom ∅ = ∅
Colors of variables: wff set class
Syntax hints:  ¬ wn 3   = wceq 1289  wex 1426  wcel 1438  c0 3284  cop 3444  dom cdm 4428
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-dif 2999  df-un 3001  df-nul 3285  df-sn 3447  df-pr 3448  df-op 3450  df-br 3838  df-dm 4438
This theorem is referenced by:  rn0  4677  fn0  5119  f0dom0  5188  f1o00  5272  rdg0  6134  frec0g  6144
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