Theorem dm0 5915

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.

Step	Hyp	Ref	Expression
1		noel 4294	. . . 4 ⊢ ¬ ⟨𝑥, 𝑦⟩ ∈ ∅
2	1	nex 1833	. . 3 ⊢ ¬ ∃𝑦⟨𝑥, 𝑦⟩ ∈ ∅
3		vex 3462	. . . 4 ⊢ 𝑥 ∈ V
4	3	eldm2 5896	. . 3 ⊢ (𝑥 ∈ dom ∅ ↔ ∃𝑦⟨𝑥, 𝑦⟩ ∈ ∅)
5	2, 4	mtbir 326	. 2 ⊢ ¬ 𝑥 ∈ dom ∅
6	5	nel0 4312	1 ⊢ dom ∅ = ∅

This proof depends on syntax axioms:   = wceq 1570  ∃wex 1812   ∈ wcel 2146  ∅c0 4289  ⟨cop 4600  dom cdm 5666

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5115  df-dm 5676

This theorem is used by:  rn0  5921  dmxpid  5925  dmxpss  6174  fn0  6673  f0dom0  6769  f10d  6862  f1o00  6863  0fv  6929  1stval  7997  bropopvvv  8094  bropfvvvv  8096  supp0  8170  tz7.44lem1  8401  tz7.44-2  8403  tz7.44-3  8404  oicl  9501  oif  9502  swrd0  14720  dmtrclfv  15081  relexpdmd  15107  nulchn  18700  symgsssg  19568  symgfisg  19569  psgnunilem5  19595  dvbsss  26098  perfdvf  26099  uhgr0e  29458  uhgr0  29460  usgr0  29630  egrsubgr  29664  0grsubgr  29665  vtxdg0e  29861  eupth0  30602  dmadjrnb  32295  eldmne0  33009  of0r  33061  f1ocnt  33182  tocyccntz  33495  mbfmcst  34681  0rrv  34873  matunitlindf  38310  ismgmOLD  38542  conrel2d  44431  neicvgbex  44879  iblempty  46720  dmrnxp  49656  reldmprcof1  50200  reldmprcof2  50201  reldmlan2  50436  reldmran2  50437