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Metamath Proof Explorer

Theorem restsn

Description: The only subspace topology induced by the topology { (/) } . (Contributed by FL, 5-Jan-2009) (Revised by Mario Carneiro, 15-Dec-2013)

Ref Expression
Assertion restsn ⊢ A ∈ V → ∅ ↾ 𝑡 A = ∅

Proof

Step Hyp Ref Expression
1 sn0top ⊢ ∅ ∈ Top
2 elrest ⊢ ∅ ∈ Top ∧ A ∈ V → x ∈ ∅ ↾ 𝑡 A ↔ ∃ y ∈ ∅ x = y ∩ A
3 1 2 mpan ⊢ A ∈ V → x ∈ ∅ ↾ 𝑡 A ↔ ∃ y ∈ ∅ x = y ∩ A
4 0ex ⊢ ∅ ∈ V
5 ineq1 ⊢ y = ∅ → y ∩ A = ∅ ∩ A
6 0in ⊢ ∅ ∩ A = ∅
7 5 6 eqtrdi ⊢ y = ∅ → y ∩ A = ∅
8 7 eqeq2d ⊢ y = ∅ → x = y ∩ A ↔ x = ∅
9 4 8 rexsn ⊢ ∃ y ∈ ∅ x = y ∩ A ↔ x = ∅
10 velsn ⊢ x ∈ ∅ ↔ x = ∅
11 9 10 bitr4i ⊢ ∃ y ∈ ∅ x = y ∩ A ↔ x ∈ ∅
12 3 11 bitrdi ⊢ A ∈ V → x ∈ ∅ ↾ 𝑡 A ↔ x ∈ ∅
13 12 eqrdv ⊢ A ∈ V → ∅ ↾ 𝑡 A = ∅