Theorem dm0 5371

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 3952	. . . 4 ⊢ ¬ ⟨𝑥, 𝑦⟩ ∈ ∅
2	1	nex 1771	. . 3 ⊢ ¬ ∃𝑦⟨𝑥, 𝑦⟩ ∈ ∅
3		vex 3234	. . . 4 ⊢ 𝑥 ∈ V
4	3	eldm2 5354	. . 3 ⊢ (𝑥 ∈ dom ∅ ↔ ∃𝑦⟨𝑥, 𝑦⟩ ∈ ∅)
5	2, 4	mtbir 312	. 2 ⊢ ¬ 𝑥 ∈ dom ∅
6	5	nel0 3965	1 ⊢ dom ∅ = ∅

Syntax hints:   = wceq 1523  ∃wex 1744   ∈ wcel 2030  ∅c0 3948  ⟨cop 4216  dom cdm 5143

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-dm 5153

This theorem is referenced by:  dmxpid  5377  rn0  5409  dmxpss  5600  fn0  6049  f0dom0  6127  f10d  6208  f1o00  6209  0fv  6265  1stval  7212  bropopvvv  7300  bropfvvvv  7302  supp0  7345  tz7.44lem1  7546  tz7.44-2  7548  tz7.44-3  7549  oicl  8475  oif  8476  swrd0  13480  dmtrclfv  13803  strlemor0OLD  16015  symgsssg  17933  symgfisg  17934  psgnunilem5  17960  dvbsss  23711  perfdvf  23712  uhgr0e  26011  uhgr0  26013  usgr0  26180  egrsubgr  26214  0grsubgr  26215  vtxdg0e  26426  eupth0  27192  dmadjrnb  28893  f1ocnt  29687  mbfmcst  30449  0rrv  30641  matunitlindf  33537  ismgmOLD  33779  conrel2d  38273  neicvgbex  38727  iblempty  40499