Theorem iccen

Description: Any nontrivial closed interval is equinumerous to the unit interval. (Contributed by Mario Carneiro, 26-Jul-2014) (Revised by Mario Carneiro, 8-Sep-2015)

	Ref	Expression
Assertion	iccen	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵 ) → ( 0 [,] 1 ) ≈ ( 𝐴 [,] 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		ovex	⊢ ( 0 [,] 1 ) ∈ V
2		ovex	⊢ ( 𝐴 [,] 𝐵 ) ∈ V
3		eqid	⊢ ( 𝑥 ∈ ( 0 [,] 1 ) ↦ ( ( 𝑥 · 𝐵 ) + ( ( 1 − 𝑥 ) · 𝐴 ) ) ) = ( 𝑥 ∈ ( 0 [,] 1 ) ↦ ( ( 𝑥 · 𝐵 ) + ( ( 1 − 𝑥 ) · 𝐴 ) ) )
4	3	iccf1o	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵 ) → ( ( 𝑥 ∈ ( 0 [,] 1 ) ↦ ( ( 𝑥 · 𝐵 ) + ( ( 1 − 𝑥 ) · 𝐴 ) ) ) : ( 0 [,] 1 ) –1-1-onto→ ( 𝐴 [,] 𝐵 ) ∧ ◡ ( 𝑥 ∈ ( 0 [,] 1 ) ↦ ( ( 𝑥 · 𝐵 ) + ( ( 1 − 𝑥 ) · 𝐴 ) ) ) = ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ↦ ( ( 𝑦 − 𝐴 ) / ( 𝐵 − 𝐴 ) ) ) ) )
5	4	simpld	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵 ) → ( 𝑥 ∈ ( 0 [,] 1 ) ↦ ( ( 𝑥 · 𝐵 ) + ( ( 1 − 𝑥 ) · 𝐴 ) ) ) : ( 0 [,] 1 ) –1-1-onto→ ( 𝐴 [,] 𝐵 ) )
6		f1oen2g	⊢ ( ( ( 0 [,] 1 ) ∈ V ∧ ( 𝐴 [,] 𝐵 ) ∈ V ∧ ( 𝑥 ∈ ( 0 [,] 1 ) ↦ ( ( 𝑥 · 𝐵 ) + ( ( 1 − 𝑥 ) · 𝐴 ) ) ) : ( 0 [,] 1 ) –1-1-onto→ ( 𝐴 [,] 𝐵 ) ) → ( 0 [,] 1 ) ≈ ( 𝐴 [,] 𝐵 ) )
7	1 2 5 6	mp3an12i	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 < 𝐵 ) → ( 0 [,] 1 ) ≈ ( 𝐴 [,] 𝐵 ) )