Theorem dmsn0 2543

Description: The domain of the singleton of the empty set is empty.

Assertion

Ref	Expression
dmsn0	⊢ dom {∅} = ∅

Proof of Theorem dmsn0

Step	Hyp	Ref	Expression
1		opnz 1897	. . . . . 6 ⊢ ¬ ⟨x, y⟩ = ∅
2		opex 1893	. . . . . . 7 ⊢ ⟨x, y⟩ ∈ V
3	2	elsnc 1826	. . . . . 6 ⊢ (⟨x, y⟩ ∈ {∅} ↔ ⟨x, y⟩ = ∅)
4	1, 3	mtbir 167	. . . . 5 ⊢ ¬ ⟨x, y⟩ ∈ {∅}
5	4	nex 779	. . . 4 ⊢ ¬ ∃y⟨x, y⟩ ∈ {∅}
6		cleqid 1102	. . . . 5 ⊢ x = x
7		negb 79	. . . . 5 ⊢ (x = x → ¬ ¬ x = x)
8	6, 7	ax-mp 6	. . . 4 ⊢ ¬ ¬ x = x
9		pm5.21 502	. . . 4 ⊢ ((¬ ∃y⟨x, y⟩ ∈ {∅} ∧ ¬ ¬ x = x) → (∃y⟨x, y⟩ ∈ {∅} ↔ ¬ x = x))
10	5, 8, 9	mp2an 520	. . 3 ⊢ (∃y⟨x, y⟩ ∈ {∅} ↔ ¬ x = x)
11	10	biabi 1181	. 2 ⊢ {x∣∃y⟨x, y⟩ ∈ {∅}} = {x∣ ¬ x = x}
12		dfdm3 2522	. 2 ⊢ dom {∅} = {x∣∃y⟨x, y⟩ ∈ {∅}}
13		dfnul2 1709	. 2 ⊢ ∅ = {x∣ ¬ x = x}
14	11, 12, 13	3eqtr4 1126	1 ⊢ dom {∅} = ∅

Syntax hints:  ¬ wn 1   ↔ wb 127  ∃wex 678   = weq 797  {cab 1090   = wceq 1091   ∈ wcel 1092  ∅c0 1707  {csn 1808  ⟨cop 1810  dom cdm 2410

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-dm 2428

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