Theorem dm0 5339

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 3919	. . . 4 ⊢ ¬ ⟨𝑥, 𝑦⟩ ∈ ∅
2	1	nex 1731	. . 3 ⊢ ¬ ∃𝑦⟨𝑥, 𝑦⟩ ∈ ∅
3		vex 3203	. . . 4 ⊢ 𝑥 ∈ V
4	3	eldm2 5322	. . 3 ⊢ (𝑥 ∈ dom ∅ ↔ ∃𝑦⟨𝑥, 𝑦⟩ ∈ ∅)
5	2, 4	mtbir 313	. 2 ⊢ ¬ 𝑥 ∈ dom ∅
6	5	nel0 3932	1 ⊢ dom ∅ = ∅

Syntax hints:   = wceq 1483  ∃wex 1704   ∈ wcel 1990  ∅c0 3915  ⟨cop 4183  dom cdm 5114

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-dm 5124

This theorem is referenced by:  dmxpid  5345  rn0  5377  dmxpss  5565  fn0  6011  f0dom0  6089  f10d  6170  f1o00  6171  0fv  6227  1stval  7170  bropopvvv  7255  bropfvvvv  7257  supp0  7300  tz7.44lem1  7501  tz7.44-2  7503  tz7.44-3  7504  oicl  8434  oif  8435  swrd0  13434  dmtrclfv  13759  strlemor0OLD  15968  symgsssg  17887  symgfisg  17888  psgnunilem5  17914  dvbsss  23666  perfdvf  23667  uhgr0e  25966  uhgr0  25968  usgr0  26135  egrsubgr  26169  0grsubgr  26170  vtxdg0e  26370  eupth0  27074  dmadjrnb  28765  f1ocnt  29559  mbfmcst  30321  0rrv  30513  matunitlindf  33407  ismgmOLD  33649  conrel2d  37956  neicvgbex  38410  iblempty  40181