Theorem dm0 4748

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 3376	. 2 ⊢ (dom ∅ = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ dom ∅)
2		noel 3362	. . . 4 ⊢ ¬ ⟨𝑥, 𝑦⟩ ∈ ∅
3	2	nex 1476	. . 3 ⊢ ¬ ∃𝑦⟨𝑥, 𝑦⟩ ∈ ∅
4		vex 2684	. . . 4 ⊢ 𝑥 ∈ V
5	4	eldm2 4732	. . 3 ⊢ (𝑥 ∈ dom ∅ ↔ ∃𝑦⟨𝑥, 𝑦⟩ ∈ ∅)
6	3, 5	mtbir 660	. 2 ⊢ ¬ 𝑥 ∈ dom ∅
7	1, 6	mpgbir 1429	1 ⊢ dom ∅ = ∅

Syntax hints:  ¬ wn 3   = wceq 1331  ∃wex 1468   ∈ wcel 1480  ∅c0 3358  ⟨cop 3525  dom cdm 4534

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-dif 3068  df-un 3070  df-nul 3359  df-sn 3528  df-pr 3529  df-op 3531  df-br 3925  df-dm 4544

This theorem is referenced by:  rn0  4790  sqxpeq0  4957  fn0  5237  f0dom0  5311  f1o00  5395  rdg0  6277  frec0g  6287  ennnfonelemj0  11903  ennnfonelem1  11909  ennnfonelemkh  11914  ennnfonelemhf1o  11915