Theorem dm0 4898

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 3481	. 2 ⊢ (dom ∅ = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ dom ∅)
2		noel 3466	. . . 4 ⊢ ¬ ⟨𝑥, 𝑦⟩ ∈ ∅
3	2	nex 1524	. . 3 ⊢ ¬ ∃𝑦⟨𝑥, 𝑦⟩ ∈ ∅
4		vex 2776	. . . 4 ⊢ 𝑥 ∈ V
5	4	eldm2 4882	. . 3 ⊢ (𝑥 ∈ dom ∅ ↔ ∃𝑦⟨𝑥, 𝑦⟩ ∈ ∅)
6	3, 5	mtbir 673	. 2 ⊢ ¬ 𝑥 ∈ dom ∅
7	1, 6	mpgbir 1477	1 ⊢ dom ∅ = ∅

Syntax hints:  ¬ wn 3   = wceq 1373  ∃wex 1516   ∈ wcel 2177  ∅c0 3462  ⟨cop 3638  dom cdm 4680

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775  df-dif 3170  df-un 3172  df-nul 3463  df-sn 3641  df-pr 3642  df-op 3644  df-br 4049  df-dm 4690

This theorem is referenced by:  rn0  4940  sqxpeq0  5112  fn0  5402  f0dom0  5478  f10d  5566  f1o00  5567  rdg0  6483  frec0g  6493  swrd0g  11127  ennnfonelemj0  12822  ennnfonelem1  12828  ennnfonelemkh  12833  ennnfonelemhf1o  12834  uhgr0e  15728  uhgr0  15731