Proof of The Borel Sets of ℝⁿ Form a σ-Algebra, and Intervals of ℝ Are Sets

Proof of (i). By Borel Sigma-Algebra of a Metric Space (iii) and Borel Sigma-Algebra of a Metric Space (i),

$$\mathcal{B}(\mathbb{R}^{n})=\sigma(\mathcal{T}_{d_{E}}),$$

with $\mathcal{T}_{d_{E}}$ a set of subsets of $\mathbb{R}^{n}$ by The Open Sets of a Metric Space Form a Set, Given by a Predicative Formula (ii), hence a subset of the power set of $\mathbb{R}^{n}$ (and $\mathbb{R}^{n}$ is a set by Euclidean Space $\mathbb{R}^n$: n-Tuples of Real Numbers with Their Componentwise Operations (i)). So $\mathcal{B}(\mathbb{R}^{n})$ is a $\sigma$-algebra on $\mathbb{R}^{n}$ by Closure Properties of σ-Algebras, Intersections and Traces of σ-Algebras, and the Generated σ-Algebra (iii), and $(\mathbb{R}^{n},\mathcal{B}(\mathbb{R}^{n}))$ is a measurable space by σ-Algebra and Measurable Space (ii).

Proof of (ii). By Intervals of the Real Line (i), $I$ is a subclass of $\mathbb{R}$, which is a set by The Real Numbers (i); hence $I$ is a set by Subclasses of Sets Are Sets, the Union and Power Set of a Set Exist Uniquely, Binary Unions of Sets Are Sets, and the Universal Class Is Proper (i). $\blacksquare$