Theorem dm0 5790

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 4296	. . . 4 ⊢ ¬ ⟨𝑥, 𝑦⟩ ∈ ∅
2	1	nex 1801	. . 3 ⊢ ¬ ∃𝑦⟨𝑥, 𝑦⟩ ∈ ∅
3		vex 3497	. . . 4 ⊢ 𝑥 ∈ V
4	3	eldm2 5770	. . 3 ⊢ (𝑥 ∈ dom ∅ ↔ ∃𝑦⟨𝑥, 𝑦⟩ ∈ ∅)
5	2, 4	mtbir 325	. 2 ⊢ ¬ 𝑥 ∈ dom ∅
6	5	nel0 4311	1 ⊢ dom ∅ = ∅

Syntax hints:   = wceq 1537  ∃wex 1780   ∈ wcel 2114  ∅c0 4291  ⟨cop 4573  dom cdm 5555

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rab 3147  df-v 3496  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-br 5067  df-dm 5565

This theorem is referenced by:  rn0  5796  dmxpid  5800  dmxpss  6028  fn0  6479  f0dom0  6563  f10d  6648  f1o00  6649  0fv  6709  1stval  7691  bropopvvv  7785  bropfvvvv  7787  supp0  7835  tz7.44lem1  8041  tz7.44-2  8043  tz7.44-3  8044  oicl  8993  oif  8994  swrd0  14020  dmtrclfv  14378  symgsssg  18595  symgfisg  18596  psgnunilem5  18622  dvbsss  24500  perfdvf  24501  uhgr0e  26856  uhgr0  26858  usgr0  27025  egrsubgr  27059  0grsubgr  27060  vtxdg0e  27256  eupth0  27993  dmadjrnb  29683  eldmne0  30373  f1ocnt  30525  tocyccntz  30786  mbfmcst  31517  0rrv  31709  matunitlindf  34905  ismgmOLD  35143  conrel2d  40058  neicvgbex  40511  iblempty  42299