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Theorem dm0 4691
 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 3328 . 2 (dom ∅ = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ dom ∅)
2 noel 3314 . . . 4 ¬ ⟨𝑥, 𝑦⟩ ∈ ∅
32nex 1444 . . 3 ¬ ∃𝑦𝑥, 𝑦⟩ ∈ ∅
4 vex 2644 . . . 4 𝑥 ∈ V
54eldm2 4675 . . 3 (𝑥 ∈ dom ∅ ↔ ∃𝑦𝑥, 𝑦⟩ ∈ ∅)
63, 5mtbir 637 . 2 ¬ 𝑥 ∈ dom ∅
71, 6mpgbir 1397 1 dom ∅ = ∅
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   = wceq 1299  ∃wex 1436   ∈ wcel 1448  ∅c0 3310  ⟨cop 3477  dom cdm 4477 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082 This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-v 2643  df-dif 3023  df-un 3025  df-nul 3311  df-sn 3480  df-pr 3481  df-op 3483  df-br 3876  df-dm 4487 This theorem is referenced by:  rn0  4731  sqxpeq0  4898  fn0  5178  f0dom0  5252  f1o00  5336  rdg0  6214  frec0g  6224  ennnfonelemj0  11706  ennnfonelem1  11712  ennnfonelemkh  11717  ennnfonelemhf1o  11718
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