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Metamath Proof Explorer

Theorem mnflt

Description: Minus infinity is less than any (finite) real. (Contributed by NM, 14-Oct-2005)

Ref Expression
Assertion mnflt ( 𝐴 ∈ ℝ → -∞ < 𝐴 )

Proof

Step Hyp Ref Expression
1 eqid ⊢ -∞ = -∞
2 olc ⊢ ( ( -∞ = -∞ ∧ 𝐴 ∈ ℝ ) → ( ( -∞ ∈ ℝ ∧ 𝐴 = +∞ ) ∨ ( -∞ = -∞ ∧ 𝐴 ∈ ℝ ) ) )
3 1 2 mpan ⊢ ( 𝐴 ∈ ℝ → ( ( -∞ ∈ ℝ ∧ 𝐴 = +∞ ) ∨ ( -∞ = -∞ ∧ 𝐴 ∈ ℝ ) ) )
4 3 olcd ⊢ ( 𝐴 ∈ ℝ → ( ( ( ( -∞ ∈ ℝ ∧ 𝐴 ∈ ℝ ) ∧ -∞ <ℝ 𝐴 ) ∨ ( -∞ = -∞ ∧ 𝐴 = +∞ ) ) ∨ ( ( -∞ ∈ ℝ ∧ 𝐴 = +∞ ) ∨ ( -∞ = -∞ ∧ 𝐴 ∈ ℝ ) ) ) )
5 mnfxr ⊢ -∞ ∈ ℝ*
6 rexr ⊢ ( 𝐴 ∈ ℝ → 𝐴 ∈ ℝ* )
7 ltxr ⊢ ( ( -∞ ∈ ℝ* ∧ 𝐴 ∈ ℝ* ) → ( -∞ < 𝐴 ↔ ( ( ( ( -∞ ∈ ℝ ∧ 𝐴 ∈ ℝ ) ∧ -∞ <ℝ 𝐴 ) ∨ ( -∞ = -∞ ∧ 𝐴 = +∞ ) ) ∨ ( ( -∞ ∈ ℝ ∧ 𝐴 = +∞ ) ∨ ( -∞ = -∞ ∧ 𝐴 ∈ ℝ ) ) ) ) )
8 5 6 7 sylancr ⊢ ( 𝐴 ∈ ℝ → ( -∞ < 𝐴 ↔ ( ( ( ( -∞ ∈ ℝ ∧ 𝐴 ∈ ℝ ) ∧ -∞ <ℝ 𝐴 ) ∨ ( -∞ = -∞ ∧ 𝐴 = +∞ ) ) ∨ ( ( -∞ ∈ ℝ ∧ 𝐴 = +∞ ) ∨ ( -∞ = -∞ ∧ 𝐴 ∈ ℝ ) ) ) ) )
9 4 8 mpbird ⊢ ( 𝐴 ∈ ℝ → -∞ < 𝐴 )