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Theorem dm0 4611
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 3287 . 2 (dom ∅ = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ dom ∅)
2 noel 3276 . . . 4 ¬ ⟨𝑥, 𝑦⟩ ∈ ∅
32nex 1432 . . 3 ¬ ∃𝑦𝑥, 𝑦⟩ ∈ ∅
4 vex 2617 . . . 4 𝑥 ∈ V
54eldm2 4595 . . 3 (𝑥 ∈ dom ∅ ↔ ∃𝑦𝑥, 𝑦⟩ ∈ ∅)
63, 5mtbir 629 . 2 ¬ 𝑥 ∈ dom ∅
71, 6mpgbir 1385 1 dom ∅ = ∅
Colors of variables: wff set class
Syntax hints:  ¬ wn 3   = wceq 1287  wex 1424  wcel 1436  c0 3272  cop 3428  dom cdm 4404
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 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-v 2616  df-dif 2988  df-un 2990  df-nul 3273  df-sn 3431  df-pr 3432  df-op 3434  df-br 3815  df-dm 4414
This theorem is referenced by:  rn0  4650  fn0  5089  f0dom0  5155  f1o00  5239  rdg0  6087  frec0g  6097
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