Theorem dm0 4975

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 3531	. 2 ⊢ (dom ∅ = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ dom ∅)
2		noel 3516	. . . 4 ⊢ ¬ ⟨𝑥, 𝑦⟩ ∈ ∅
3	2	nex 1549	. . 3 ⊢ ¬ ∃𝑦⟨𝑥, 𝑦⟩ ∈ ∅
4		vex 2818	. . . 4 ⊢ 𝑥 ∈ V
5	4	eldm2 4959	. . 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 2205  ∅c0 3512  ⟨cop 3697  dom cdm 4754

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 2216

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 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817  df-dif 3216  df-un 3218  df-nul 3513  df-sn 3700  df-pr 3701  df-op 3703  df-br 4115  df-dm 4764

This theorem is referenced by:  rn0  5018  sqxpeq0  5191  fn0  5483  f0dom0  5566  f10d  5655  f1o00  5656  supp0  6451  rdg0  6631  frec0g  6641  swrd0g  11380  ennnfonelemj0  13240  ennnfonelem1  13246  ennnfonelemkh  13251  ennnfonelemhf1o  13252  uhgr0e  16207  uhgr0  16210  usgr0  16364  egrsubgr  16388  0grsubgr  16389  vtxdgfi0e  16420