This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem ltpnf

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

Ref Expression
Assertion ltpnf ( 𝐴 ∈ ℝ → 𝐴 < +∞ )

Proof

Step Hyp Ref Expression
1 eqid ⊢ +∞ = +∞
2 orc ⊢ ( ( 𝐴 ∈ ℝ ∧ +∞ = +∞ ) → ( ( 𝐴 ∈ ℝ ∧ +∞ = +∞ ) ∨ ( 𝐴 = -∞ ∧ +∞ ∈ ℝ ) ) )
3 1 2 mpan2 ⊢ ( 𝐴 ∈ ℝ → ( ( 𝐴 ∈ ℝ ∧ +∞ = +∞ ) ∨ ( 𝐴 = -∞ ∧ +∞ ∈ ℝ ) ) )
4 3 olcd ⊢ ( 𝐴 ∈ ℝ → ( ( ( ( 𝐴 ∈ ℝ ∧ +∞ ∈ ℝ ) ∧ 𝐴 <ℝ +∞ ) ∨ ( 𝐴 = -∞ ∧ +∞ = +∞ ) ) ∨ ( ( 𝐴 ∈ ℝ ∧ +∞ = +∞ ) ∨ ( 𝐴 = -∞ ∧ +∞ ∈ ℝ ) ) ) )
5 rexr ⊢ ( 𝐴 ∈ ℝ → 𝐴 ∈ ℝ* )
6 pnfxr ⊢ +∞ ∈ ℝ*
7 ltxr ⊢ ( ( 𝐴 ∈ ℝ* ∧ +∞ ∈ ℝ* ) → ( 𝐴 < +∞ ↔ ( ( ( ( 𝐴 ∈ ℝ ∧ +∞ ∈ ℝ ) ∧ 𝐴 <ℝ +∞ ) ∨ ( 𝐴 = -∞ ∧ +∞ = +∞ ) ) ∨ ( ( 𝐴 ∈ ℝ ∧ +∞ = +∞ ) ∨ ( 𝐴 = -∞ ∧ +∞ ∈ ℝ ) ) ) ) )
8 5 6 7 sylancl ⊢ ( 𝐴 ∈ ℝ → ( 𝐴 < +∞ ↔ ( ( ( ( 𝐴 ∈ ℝ ∧ +∞ ∈ ℝ ) ∧ 𝐴 <ℝ +∞ ) ∨ ( 𝐴 = -∞ ∧ +∞ = +∞ ) ) ∨ ( ( 𝐴 ∈ ℝ ∧ +∞ = +∞ ) ∨ ( 𝐴 = -∞ ∧ +∞ ∈ ℝ ) ) ) ) )
9 4 8 mpbird ⊢ ( 𝐴 ∈ ℝ → 𝐴 < +∞ )