Theorem dm0 4934

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 3510	. 2 ⊢ (dom ∅ = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ dom ∅)
2		noel 3495	. . . 4 ⊢ ¬ ⟨𝑥, 𝑦⟩ ∈ ∅
3	2	nex 1546	. . 3 ⊢ ¬ ∃𝑦⟨𝑥, 𝑦⟩ ∈ ∅
4		vex 2802	. . . 4 ⊢ 𝑥 ∈ V
5	4	eldm2 4918	. . 3 ⊢ (𝑥 ∈ dom ∅ ↔ ∃𝑦⟨𝑥, 𝑦⟩ ∈ ∅)
6	3, 5	mtbir 675	. 2 ⊢ ¬ 𝑥 ∈ dom ∅
7	1, 6	mpgbir 1499	1 ⊢ dom ∅ = ∅

Syntax hints:  ¬ wn 3   = wceq 1395  ∃wex 1538   ∈ wcel 2200  ∅c0 3491  ⟨cop 3669  dom cdm 4716

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-dif 3199  df-un 3201  df-nul 3492  df-sn 3672  df-pr 3673  df-op 3675  df-br 4083  df-dm 4726

This theorem is referenced by:  rn0  4976  sqxpeq0  5148  fn0  5439  f0dom0  5515  f10d  5603  f1o00  5604  rdg0  6523  frec0g  6533  swrd0g  11178  ennnfonelemj0  12958  ennnfonelem1  12964  ennnfonelemkh  12969  ennnfonelemhf1o  12970  uhgr0e  15867  uhgr0  15870