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Theorem dm0 5876
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 4290 . . . 4 ¬ ⟨𝑥, 𝑦⟩ ∈ ∅
21nex 1802 . . 3 ¬ ∃𝑦𝑥, 𝑦⟩ ∈ ∅
3 vex 3449 . . . 4 𝑥 ∈ V
43eldm2 5857 . . 3 (𝑥 ∈ dom ∅ ↔ ∃𝑦𝑥, 𝑦⟩ ∈ ∅)
52, 4mtbir 322 . 2 ¬ 𝑥 ∈ dom ∅
65nel0 4310 1 dom ∅ = ∅
Colors of variables: wff setvar class
Syntax hints:   = wceq 1541  wex 1781  wcel 2106  c0 4282  cop 4592  dom cdm 5633
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-br 5106  df-dm 5643
This theorem is referenced by:  rn0  5881  dmxpid  5885  dmxpss  6123  fn0  6632  f0dom0  6726  f10d  6818  f1o00  6819  0fv  6886  1stval  7922  bropopvvv  8021  bropfvvvv  8023  supp0  8096  tz7.44lem1  8350  tz7.44-2  8352  tz7.44-3  8353  oicl  9464  oif  9465  swrd0  14545  dmtrclfv  14902  relexpdmd  14928  symgsssg  19247  symgfisg  19248  psgnunilem5  19274  dvbsss  25264  perfdvf  25265  uhgr0e  28020  uhgr0  28022  usgr0  28189  egrsubgr  28223  0grsubgr  28224  vtxdg0e  28420  eupth0  29156  dmadjrnb  30846  eldmne0  31540  f1ocnt  31700  tocyccntz  31988  mbfmcst  32850  0rrv  33042  matunitlindf  36067  ismgmOLD  36300  conrel2d  41918  neicvgbex  42366  iblempty  44178
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