Theorem dm0 5869

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 4279	. . . 4 ⊢ ¬ ⟨𝑥, 𝑦⟩ ∈ ∅
2	1	nex 1802	. . 3 ⊢ ¬ ∃𝑦⟨𝑥, 𝑦⟩ ∈ ∅
3		vex 3434	. . . 4 ⊢ 𝑥 ∈ V
4	3	eldm2 5850	. . 3 ⊢ (𝑥 ∈ dom ∅ ↔ ∃𝑦⟨𝑥, 𝑦⟩ ∈ ∅)
5	2, 4	mtbir 323	. 2 ⊢ ¬ 𝑥 ∈ dom ∅
6	5	nel0 4295	1 ⊢ dom ∅ = ∅

Syntax hints:   = wceq 1542  ∃wex 1781   ∈ wcel 2114  ∅c0 4274  ⟨cop 4574  dom cdm 5624

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-dm 5634

This theorem is referenced by:  rn0  5875  dmxpid  5879  dmxpss  6129  fn0  6623  f0dom0  6718  f10d  6808  f1o00  6809  0fv  6875  1stval  7937  bropopvvv  8033  bropfvvvv  8035  supp0  8108  tz7.44lem1  8337  tz7.44-2  8339  tz7.44-3  8340  oicl  9437  oif  9438  swrd0  14612  dmtrclfv  14971  relexpdmd  14997  nulchn  18576  symgsssg  19433  symgfisg  19434  psgnunilem5  19460  dvbsss  25879  perfdvf  25880  uhgr0e  29154  uhgr0  29156  usgr0  29326  egrsubgr  29360  0grsubgr  29361  vtxdg0e  29558  eupth0  30299  dmadjrnb  31992  eldmne0  32715  of0r  32767  f1ocnt  32888  tocyccntz  33220  mbfmcst  34419  0rrv  34611  matunitlindf  37953  ismgmOLD  38185  conrel2d  44109  neicvgbex  44557  iblempty  46411  dmrnxp  49324  reldmprcof1  49868  reldmprcof2  49869  reldmlan2  50104  reldmran2  50105