Theorem dm0 4969

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 3526	. 2 ⊢ (dom ∅ = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ dom ∅)
2		noel 3511	. . . 4 ⊢ ¬ ⟨𝑥, 𝑦⟩ ∈ ∅
3	2	nex 1549	. . 3 ⊢ ¬ ∃𝑦⟨𝑥, 𝑦⟩ ∈ ∅
4		vex 2815	. . . 4 ⊢ 𝑥 ∈ V
5	4	eldm2 4953	. . 3 ⊢ (𝑥 ∈ dom ∅ ↔ ∃𝑦⟨𝑥, 𝑦⟩ ∈ ∅)
6	3, 5	mtbir 678	. 2 ⊢ ¬ 𝑥 ∈ dom ∅
7	1, 6	mpgbir 1502	1 ⊢ dom ∅ = ∅

Syntax hints:  ¬ wn 3   = wceq 1398  ∃wex 1541   ∈ wcel 2203  ∅c0 3507  ⟨cop 3691  dom cdm 4748

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2814  df-dif 3212  df-un 3214  df-nul 3508  df-sn 3694  df-pr 3695  df-op 3697  df-br 4109  df-dm 4758

This theorem is referenced by:  rn0  5012  sqxpeq0  5185  fn0  5477  f0dom0  5560  f10d  5649  f1o00  5650  supp0  6437  rdg0  6617  frec0g  6627  swrd0g  11348  ennnfonelemj0  13144  ennnfonelem1  13150  ennnfonelemkh  13155  ennnfonelemhf1o  13156  uhgr0e  16069  uhgr0  16072  usgr0  16226  egrsubgr  16250  0grsubgr  16251  vtxdgfi0e  16282