Theorem dmsnsn0 2544

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

Assertion

Ref	Expression
dmsnsn0	⊢ dom {{∅}} = ∅

Proof of Theorem dmsnsn0

Step	Hyp	Ref	Expression
1		visset 1350	. . . . . . . . . 10 ⊢ y ∈ V
2	1	a1i 7	. . . . . . . . 9 ⊢ (¬ x ∈ V → y ∈ V)
3	2	orri 201	. . . . . . . 8 ⊢ (x ∈ V ∨ y ∈ V)
4		oran 255	. . . . . . . 8 ⊢ ((x ∈ V ∨ y ∈ V) ↔ ¬ (¬ x ∈ V ∧ ¬ y ∈ V))
5	3, 4	mpbi 164	. . . . . . 7 ⊢ ¬ (¬ x ∈ V ∧ ¬ y ∈ V)
6		opprc3 1908	. . . . . . 7 ⊢ ((¬ x ∈ V ∧ ¬ y ∈ V) ↔ ⟨x, y⟩ = {∅})
7	5, 6	mtbi 166	. . . . . 6 ⊢ ¬ ⟨x, y⟩ = {∅}
8		opex 1893	. . . . . . 7 ⊢ ⟨x, y⟩ ∈ V
9	8	elsnc 1826	. . . . . 6 ⊢ (⟨x, y⟩ ∈ {{∅}} ↔ ⟨x, y⟩ = {∅})
10	7, 9	mtbir 167	. . . . 5 ⊢ ¬ ⟨x, y⟩ ∈ {{∅}}
11	10	nex 779	. . . 4 ⊢ ¬ ∃y⟨x, y⟩ ∈ {{∅}}
12		cleqid 1102	. . . . 5 ⊢ x = x
13		negb 79	. . . . 5 ⊢ (x = x → ¬ ¬ x = x)
14	12, 13	ax-mp 6	. . . 4 ⊢ ¬ ¬ x = x
15		pm5.21 502	. . . 4 ⊢ ((¬ ∃y⟨x, y⟩ ∈ {{∅}} ∧ ¬ ¬ x = x) → (∃y⟨x, y⟩ ∈ {{∅}} ↔ ¬ x = x))
16	11, 14, 15	mp2an 520	. . 3 ⊢ (∃y⟨x, y⟩ ∈ {{∅}} ↔ ¬ x = x)
17	16	biabi 1181	. 2 ⊢ {x∣∃y⟨x, y⟩ ∈ {{∅}}} = {x∣ ¬ x = x}
18		dfdm3 2522	. 2 ⊢ dom {{∅}} = {x∣∃y⟨x, y⟩ ∈ {{∅}}}
19		dfnul2 1709	. 2 ⊢ ∅ = {x∣ ¬ x = x}
20	17, 18, 19	3eqtr4 1126	1 ⊢ dom {{∅}} = ∅

Syntax hints:  ¬ wn 1   ↔ wb 127   ∨ wo 195   ∧ wa 196  ∃wex 678   = weq 797  {cab 1090   = wceq 1091   ∈ wcel 1092  Vcvv 1348  ∅c0 1707  {csn 1808  ⟨cop 1810  dom cdm 2410

This theorem is referenced by:  dmsnop 2547

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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