Theorem ioopos

Description: The set of positive reals expressed as an open interval. (Contributed by NM, 7-May-2007)

	Ref	Expression
Assertion	ioopos	⊢ ( 0 (,) +∞ ) = { 𝑥 ∈ ℝ ∣ 0 < 𝑥 }

Proof

Step	Hyp	Ref	Expression
1		0xr	⊢ 0 ∈ ℝ*
2		pnfxr	⊢ +∞ ∈ ℝ*
3		iooval2	⊢ ( ( 0 ∈ ℝ* ∧ +∞ ∈ ℝ* ) → ( 0 (,) +∞ ) = { 𝑥 ∈ ℝ ∣ ( 0 < 𝑥 ∧ 𝑥 < +∞ ) } )
4	1 2 3	mp2an	⊢ ( 0 (,) +∞ ) = { 𝑥 ∈ ℝ ∣ ( 0 < 𝑥 ∧ 𝑥 < +∞ ) }
5		ltpnf	⊢ ( 𝑥 ∈ ℝ → 𝑥 < +∞ )
6	5	biantrud	⊢ ( 𝑥 ∈ ℝ → ( 0 < 𝑥 ↔ ( 0 < 𝑥 ∧ 𝑥 < +∞ ) ) )
7	6	rabbiia	⊢ { 𝑥 ∈ ℝ ∣ 0 < 𝑥 } = { 𝑥 ∈ ℝ ∣ ( 0 < 𝑥 ∧ 𝑥 < +∞ ) }
8	4 7	eqtr4i	⊢ ( 0 (,) +∞ ) = { 𝑥 ∈ ℝ ∣ 0 < 𝑥 }