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Theorem dm0 5784
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 noel 4295 . . . 4 ¬ ⟨𝑥, 𝑦⟩ ∈ ∅
21nex 1792 . . 3 ¬ ∃𝑦𝑥, 𝑦⟩ ∈ ∅
3 vex 3498 . . . 4 𝑥 ∈ V
43eldm2 5764 . . 3 (𝑥 ∈ dom ∅ ↔ ∃𝑦𝑥, 𝑦⟩ ∈ ∅)
52, 4mtbir 324 . 2 ¬ 𝑥 ∈ dom ∅
65nel0 4310 1 dom ∅ = ∅
Colors of variables: wff setvar class
Syntax hints:   = wceq 1528  wex 1771  wcel 2105  c0 4290  cop 4565  dom cdm 5549
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2793
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rab 3147  df-v 3497  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-br 5059  df-dm 5559
This theorem is referenced by:  rn0  5790  dmxpid  5794  dmxpss  6022  fn0  6473  f0dom0  6557  f10d  6642  f1o00  6643  0fv  6703  1stval  7682  bropopvvv  7776  bropfvvvv  7778  supp0  7826  tz7.44lem1  8032  tz7.44-2  8034  tz7.44-3  8035  oicl  8982  oif  8983  swrd0  14010  dmtrclfv  14368  symgsssg  18526  symgfisg  18527  psgnunilem5  18553  dvbsss  24429  perfdvf  24430  uhgr0e  26784  uhgr0  26786  usgr0  26953  egrsubgr  26987  0grsubgr  26988  vtxdg0e  27184  eupth0  27921  dmadjrnb  29611  eldmne0  30302  f1ocnt  30452  tocyccntz  30714  mbfmcst  31417  0rrv  31609  matunitlindf  34772  ismgmOLD  35011  conrel2d  39889  neicvgbex  40342  iblempty  42130
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