Theorem dm0 4995

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 4979	. . 3 ⊢ (𝑥 ∈ dom ∅ ↔ ∃𝑦⟨𝑥, 𝑦⟩ ∈ ∅)
6	3, 5	mtbir 682	. 2 ⊢ ¬ 𝑥 ∈ dom ∅
7	1, 6	mpgbir 1506	1 ⊢ dom ∅ = ∅

This proof depends on syntax axioms:  ¬ wn 3   = wceq 1402  ∃wex 1545   ∈ wcel 2209  ∅c0 3520  ⟨cop 3712  dom cdm 4774

This proof depends on 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 proof 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 3715  df-pr 3716  df-op 3718  df-br 4131  df-dm 4784

This theorem is used by:  rn0  5038  sqxpeq0  5211  fn0  5503  f0dom0  5586  f10d  5675  f1o00  5676  supp0  6478  rdg0  6658  frec0g  6668  swrd0g  11434  ennnfonelemj0  13294  ennnfonelem1  13300  ennnfonelemkh  13305  ennnfonelemhf1o  13306  uhgr0e  16335  uhgr0  16338  usgr0  16492  egrsubgr  16516  0grsubgr  16517  vtxdgfi0e  16548