Theorem dm0 4881

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 3470	. 2 ⊢ (dom ∅ = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ dom ∅)
2		noel 3455	. . . 4 ⊢ ¬ ⟨𝑥, 𝑦⟩ ∈ ∅
3	2	nex 1514	. . 3 ⊢ ¬ ∃𝑦⟨𝑥, 𝑦⟩ ∈ ∅
4		vex 2766	. . . 4 ⊢ 𝑥 ∈ V
5	4	eldm2 4865	. . 3 ⊢ (𝑥 ∈ dom ∅ ↔ ∃𝑦⟨𝑥, 𝑦⟩ ∈ ∅)
6	3, 5	mtbir 672	. 2 ⊢ ¬ 𝑥 ∈ dom ∅
7	1, 6	mpgbir 1467	1 ⊢ dom ∅ = ∅

Syntax hints:  ¬ wn 3   = wceq 1364  ∃wex 1506   ∈ wcel 2167  ∅c0 3451  ⟨cop 3626  dom cdm 4664

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-dif 3159  df-un 3161  df-nul 3452  df-sn 3629  df-pr 3630  df-op 3632  df-br 4035  df-dm 4674

This theorem is referenced by:  rn0  4923  sqxpeq0  5094  fn0  5380  f0dom0  5454  f1o00  5542  rdg0  6454  frec0g  6464  ennnfonelemj0  12643  ennnfonelem1  12649  ennnfonelemkh  12654  ennnfonelemhf1o  12655