Theorem dm0 4940

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 4924	. . 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 4720

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 4084  df-dm 4730

This theorem is referenced by:  rn0  4983  sqxpeq0  5155  fn0  5446  f0dom0  5524  f10d  5612  f1o00  5613  rdg0  6544  frec0g  6554  swrd0g  11213  ennnfonelemj0  12993  ennnfonelem1  12999  ennnfonelemkh  13004  ennnfonelemhf1o  13005  uhgr0e  15903  uhgr0  15906  usgr0  16058  vtxdgfi0e  16081