Description: The limit of a Hilbert space sequence is unique. (Contributed by NM, 19-Aug-1999) (Revised by Mario Carneiro, 14-May-2014) (New usage is discouraged.)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | hlimreui | ⊢ ( ∃ 𝑥 ∈ 𝐻 𝐹 ⇝𝑣 𝑥 ↔ ∃! 𝑥 ∈ 𝐻 𝐹 ⇝𝑣 𝑥 ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | hlimuni | ⊢ ( ( 𝐹 ⇝𝑣 𝑥 ∧ 𝐹 ⇝𝑣 𝑦 ) → 𝑥 = 𝑦 ) | |
| 2 | 1 | rgen2w | ⊢ ∀ 𝑥 ∈ 𝐻 ∀ 𝑦 ∈ 𝐻 ( ( 𝐹 ⇝𝑣 𝑥 ∧ 𝐹 ⇝𝑣 𝑦 ) → 𝑥 = 𝑦 ) |
| 3 | 2 | biantru | ⊢ ( ∃ 𝑥 ∈ 𝐻 𝐹 ⇝𝑣 𝑥 ↔ ( ∃ 𝑥 ∈ 𝐻 𝐹 ⇝𝑣 𝑥 ∧ ∀ 𝑥 ∈ 𝐻 ∀ 𝑦 ∈ 𝐻 ( ( 𝐹 ⇝𝑣 𝑥 ∧ 𝐹 ⇝𝑣 𝑦 ) → 𝑥 = 𝑦 ) ) ) |
| 4 | breq2 | ⊢ ( 𝑥 = 𝑦 → ( 𝐹 ⇝𝑣 𝑥 ↔ 𝐹 ⇝𝑣 𝑦 ) ) | |
| 5 | 4 | reu4 | ⊢ ( ∃! 𝑥 ∈ 𝐻 𝐹 ⇝𝑣 𝑥 ↔ ( ∃ 𝑥 ∈ 𝐻 𝐹 ⇝𝑣 𝑥 ∧ ∀ 𝑥 ∈ 𝐻 ∀ 𝑦 ∈ 𝐻 ( ( 𝐹 ⇝𝑣 𝑥 ∧ 𝐹 ⇝𝑣 𝑦 ) → 𝑥 = 𝑦 ) ) ) |
| 6 | 3 5 | bitr4i | ⊢ ( ∃ 𝑥 ∈ 𝐻 𝐹 ⇝𝑣 𝑥 ↔ ∃! 𝑥 ∈ 𝐻 𝐹 ⇝𝑣 𝑥 ) |