Theorem xnn0xrnemnf

Description: The extended nonnegative integers are extended reals without negative infinity. (Contributed by AV, 10-Dec-2020)

	Ref	Expression
Assertion	xnn0xrnemnf	⊢ ( 𝐴 ∈ ℕ0* → ( 𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞ ) )

Proof

Step	Hyp	Ref	Expression
1		xnn0xr	⊢ ( 𝐴 ∈ ℕ0* → 𝐴 ∈ ℝ* )
2		xnn0nemnf	⊢ ( 𝐴 ∈ ℕ0* → 𝐴 ≠ -∞ )
3	1 2	jca	⊢ ( 𝐴 ∈ ℕ0* → ( 𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞ ) )