Theorem dm0 4761

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 3386	. 2 ⊢ (dom ∅ = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ dom ∅)
2		noel 3372	. . . 4 ⊢ ¬ ⟨𝑥, 𝑦⟩ ∈ ∅
3	2	nex 1477	. . 3 ⊢ ¬ ∃𝑦⟨𝑥, 𝑦⟩ ∈ ∅
4		vex 2692	. . . 4 ⊢ 𝑥 ∈ V
5	4	eldm2 4745	. . 3 ⊢ (𝑥 ∈ dom ∅ ↔ ∃𝑦⟨𝑥, 𝑦⟩ ∈ ∅)
6	3, 5	mtbir 661	. 2 ⊢ ¬ 𝑥 ∈ dom ∅
7	1, 6	mpgbir 1430	1 ⊢ dom ∅ = ∅

Syntax hints:  ¬ wn 3   = wceq 1332  ∃wex 1469   ∈ wcel 1481  ∅c0 3368  ⟨cop 3535  dom cdm 4547

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-dif 3078  df-un 3080  df-nul 3369  df-sn 3538  df-pr 3539  df-op 3541  df-br 3938  df-dm 4557

This theorem is referenced by:  rn0  4803  sqxpeq0  4970  fn0  5250  f0dom0  5324  f1o00  5410  rdg0  6292  frec0g  6302  ennnfonelemj0  11950  ennnfonelem1  11956  ennnfonelemkh  11961  ennnfonelemhf1o  11962