Theorem dm0 4951

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 3515	. 2 ⊢ (dom ∅ = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ dom ∅)
2		noel 3500	. . . 4 ⊢ ¬ ⟨𝑥, 𝑦⟩ ∈ ∅
3	2	nex 1549	. . 3 ⊢ ¬ ∃𝑦⟨𝑥, 𝑦⟩ ∈ ∅
4		vex 2806	. . . 4 ⊢ 𝑥 ∈ V
5	4	eldm2 4935	. . 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 2202  ∅c0 3496  ⟨cop 3676  dom cdm 4731

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 2213

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-dif 3203  df-un 3205  df-nul 3497  df-sn 3679  df-pr 3680  df-op 3682  df-br 4094  df-dm 4741

This theorem is referenced by:  rn0  4994  sqxpeq0  5167  fn0  5459  f0dom0  5539  f10d  5628  f1o00  5629  supp0  6416  rdg0  6596  frec0g  6606  swrd0g  11288  ennnfonelemj0  13083  ennnfonelem1  13089  ennnfonelemkh  13094  ennnfonelemhf1o  13095  uhgr0e  16003  uhgr0  16006  usgr0  16160  egrsubgr  16184  0grsubgr  16185  vtxdgfi0e  16216