Theorem dm0 4993

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 3540	. 2 ⊢ (dom ∅ = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ dom ∅)
2		noel 3525	. . . 4 ⊢ ¬ ⟨𝑥, 𝑦⟩ ∈ ∅
3	2	nex 1553	. . 3 ⊢ ¬ ∃𝑦⟨𝑥, 𝑦⟩ ∈ ∅
4		vex 2824	. . . 4 ⊢ 𝑥 ∈ V
5	4	eldm2 4977	. . 3 ⊢ (𝑥 ∈ dom ∅ ↔ ∃𝑦⟨𝑥, 𝑦⟩ ∈ ∅)
6	3, 5	mtbir 682	. 2 ⊢ ¬ 𝑥 ∈ dom ∅
7	1, 6	mpgbir 1506	1 ⊢ dom ∅ = ∅

Syntax hints:  ¬ wn 3   = wceq 1402  ∃wex 1545   ∈ wcel 2209  ∅c0 3520  ⟨cop 3711  dom cdm 4772

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-dif 3222  df-un 3224  df-nul 3521  df-sn 3714  df-pr 3715  df-op 3717  df-br 4129  df-dm 4782

This theorem is referenced by:  rn0  5036  sqxpeq0  5209  fn0  5501  f0dom0  5584  f10d  5673  f1o00  5674  supp0  6472  rdg0  6652  frec0g  6662  swrd0g  11415  ennnfonelemj0  13275  ennnfonelem1  13281  ennnfonelemkh  13286  ennnfonelemhf1o  13287  uhgr0e  16306  uhgr0  16309  usgr0  16463  egrsubgr  16487  0grsubgr  16488  vtxdgfi0e  16519