Theorem mnfltxr

Description: Minus infinity is less than an extended real that is either real or plus infinity. (Contributed by NM, 2-Feb-2006)

	Ref	Expression
Assertion	mnfltxr	⊢ ( ( 𝐴 ∈ ℝ ∨ 𝐴 = +∞ ) → -∞ < 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		mnflt	⊢ ( 𝐴 ∈ ℝ → -∞ < 𝐴 )
2		mnfltpnf	⊢ -∞ < +∞
3		breq2	⊢ ( 𝐴 = +∞ → ( -∞ < 𝐴 ↔ -∞ < +∞ ) )
4	2 3	mpbiri	⊢ ( 𝐴 = +∞ → -∞ < 𝐴 )
5	1 4	jaoi	⊢ ( ( 𝐴 ∈ ℝ ∨ 𝐴 = +∞ ) → -∞ < 𝐴 )