Theorem dm0 4876

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 3465	. 2 ⊢ (dom ∅ = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ dom ∅)
2		noel 3450	. . . 4 ⊢ ¬ ⟨𝑥, 𝑦⟩ ∈ ∅
3	2	nex 1511	. . 3 ⊢ ¬ ∃𝑦⟨𝑥, 𝑦⟩ ∈ ∅
4		vex 2763	. . . 4 ⊢ 𝑥 ∈ V
5	4	eldm2 4860	. . 3 ⊢ (𝑥 ∈ dom ∅ ↔ ∃𝑦⟨𝑥, 𝑦⟩ ∈ ∅)
6	3, 5	mtbir 672	. 2 ⊢ ¬ 𝑥 ∈ dom ∅
7	1, 6	mpgbir 1464	1 ⊢ dom ∅ = ∅

Syntax hints:  ¬ wn 3   = wceq 1364  ∃wex 1503   ∈ wcel 2164  ∅c0 3446  ⟨cop 3621  dom cdm 4659

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-dif 3155  df-un 3157  df-nul 3447  df-sn 3624  df-pr 3625  df-op 3627  df-br 4030  df-dm 4669

This theorem is referenced by:  rn0  4918  sqxpeq0  5089  fn0  5373  f0dom0  5447  f1o00  5535  rdg0  6440  frec0g  6450  ennnfonelemj0  12558  ennnfonelem1  12564  ennnfonelemkh  12569  ennnfonelemhf1o  12570