Theorem dm0 5260

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		eq0 3888	. 2 ⊢ (dom ∅ = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ dom ∅)
2		noel 3878	. . . 4 ⊢ ¬ ⟨𝑥, 𝑦⟩ ∈ ∅
3	2	nex 1722	. . 3 ⊢ ¬ ∃𝑦⟨𝑥, 𝑦⟩ ∈ ∅
4		vex 3176	. . . 4 ⊢ 𝑥 ∈ V
5	4	eldm2 5244	. . 3 ⊢ (𝑥 ∈ dom ∅ ↔ ∃𝑦⟨𝑥, 𝑦⟩ ∈ ∅)
6	3, 5	mtbir 312	. 2 ⊢ ¬ 𝑥 ∈ dom ∅
7	1, 6	mpgbir 1717	1 ⊢ dom ∅ = ∅

Syntax hints:  ¬ wn 3   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ∅c0 3874  ⟨cop 4131  dom cdm 5038

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-dm 5048

This theorem is referenced by:  dmxpid  5266  rn0  5298  dmxpss  5484  fn0  5924  f0dom0  6002  f10d  6082  f1o00  6083  0fv  6137  1stval  7061  bropopvvv  7142  bropfvvvv  7144  supp0  7187  tz7.44lem1  7388  tz7.44-2  7390  tz7.44-3  7391  oicl  8317  oif  8318  swrd0  13286  dmtrclfv  13607  strlemor0  15795  symgsssg  17710  symgfisg  17711  psgnunilem5  17737  dvbsss  23472  perfdvf  23473  uhgr0e  25737  uhgr0  25739  uhgra0  25838  umgra0  25854  clwwlknprop  26300  2wlkonot3v  26402  2spthonot3v  26403  eupa0  26501  dmadjrnb  28149  f1ocnt  28946  mbfmcst  29648  0rrv  29840  matunitlindf  32577  ismgmOLD  32819  conrel2d  36975  neicvgbex  37430  iblempty  38857  usgr0  40469  egrsubgr  40501  0grsubgr  40502  vtxdg0e  40689  eupth0  41382