Theorem dm0 4843

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 3443	. 2 ⊢ (dom ∅ = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ dom ∅)
2		noel 3428	. . . 4 ⊢ ¬ ⟨𝑥, 𝑦⟩ ∈ ∅
3	2	nex 1500	. . 3 ⊢ ¬ ∃𝑦⟨𝑥, 𝑦⟩ ∈ ∅
4		vex 2742	. . . 4 ⊢ 𝑥 ∈ V
5	4	eldm2 4827	. . 3 ⊢ (𝑥 ∈ dom ∅ ↔ ∃𝑦⟨𝑥, 𝑦⟩ ∈ ∅)
6	3, 5	mtbir 671	. 2 ⊢ ¬ 𝑥 ∈ dom ∅
7	1, 6	mpgbir 1453	1 ⊢ dom ∅ = ∅

Syntax hints:  ¬ wn 3   = wceq 1353  ∃wex 1492   ∈ wcel 2148  ∅c0 3424  ⟨cop 3597  dom cdm 4628

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-dif 3133  df-un 3135  df-nul 3425  df-sn 3600  df-pr 3601  df-op 3603  df-br 4006  df-dm 4638

This theorem is referenced by:  rn0  4885  sqxpeq0  5054  fn0  5337  f0dom0  5411  f1o00  5498  rdg0  6390  frec0g  6400  ennnfonelemj0  12404  ennnfonelem1  12410  ennnfonelemkh  12415  ennnfonelemhf1o  12416