Theorem dm0 4945

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 3513	. 2 ⊢ (dom ∅ = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ dom ∅)
2		noel 3498	. . . 4 ⊢ ¬ ⟨𝑥, 𝑦⟩ ∈ ∅
3	2	nex 1548	. . 3 ⊢ ¬ ∃𝑦⟨𝑥, 𝑦⟩ ∈ ∅
4		vex 2805	. . . 4 ⊢ 𝑥 ∈ V
5	4	eldm2 4929	. . 3 ⊢ (𝑥 ∈ dom ∅ ↔ ∃𝑦⟨𝑥, 𝑦⟩ ∈ ∅)
6	3, 5	mtbir 677	. 2 ⊢ ¬ 𝑥 ∈ dom ∅
7	1, 6	mpgbir 1501	1 ⊢ dom ∅ = ∅

Syntax hints:  ¬ wn 3   = wceq 1397  ∃wex 1540   ∈ wcel 2202  ∅c0 3494  ⟨cop 3672  dom cdm 4725

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-dif 3202  df-un 3204  df-nul 3495  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-dm 4735

This theorem is referenced by:  rn0  4988  sqxpeq0  5160  fn0  5452  f0dom0  5530  f10d  5619  f1o00  5620  rdg0  6553  frec0g  6563  swrd0g  11245  ennnfonelemj0  13027  ennnfonelem1  13033  ennnfonelemkh  13038  ennnfonelemhf1o  13039  uhgr0e  15939  uhgr0  15942  usgr0  16096  egrsubgr  16120  0grsubgr  16121  vtxdgfi0e  16152