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Metamath Proof Explorer

Theorem pnfnlt

Description: No extended real is greater than plus infinity. (Contributed by NM, 15-Oct-2005)

Ref Expression
Assertion pnfnlt ( 𝐴 ∈ ℝ* → ¬ +∞ < 𝐴 )

Proof

Step Hyp Ref Expression
1 pnfnre ⊢ +∞ ∉ ℝ
2 1 neli ⊢ ¬ +∞ ∈ ℝ
3 2 intnanr ⊢ ¬ ( +∞ ∈ ℝ ∧ 𝐴 ∈ ℝ )
4 3 intnanr ⊢ ¬ ( ( +∞ ∈ ℝ ∧ 𝐴 ∈ ℝ ) ∧ +∞ <ℝ 𝐴 )
5 pnfnemnf ⊢ +∞ ≠ -∞
6 5 neii ⊢ ¬ +∞ = -∞
7 6 intnanr ⊢ ¬ ( +∞ = -∞ ∧ 𝐴 = +∞ )
8 4 7 pm3.2ni ⊢ ¬ ( ( ( +∞ ∈ ℝ ∧ 𝐴 ∈ ℝ ) ∧ +∞ <ℝ 𝐴 ) ∨ ( +∞ = -∞ ∧ 𝐴 = +∞ ) )
9 2 intnanr ⊢ ¬ ( +∞ ∈ ℝ ∧ 𝐴 = +∞ )
10 6 intnanr ⊢ ¬ ( +∞ = -∞ ∧ 𝐴 ∈ ℝ )
11 9 10 pm3.2ni ⊢ ¬ ( ( +∞ ∈ ℝ ∧ 𝐴 = +∞ ) ∨ ( +∞ = -∞ ∧ 𝐴 ∈ ℝ ) )
12 8 11 pm3.2ni ⊢ ¬ ( ( ( ( +∞ ∈ ℝ ∧ 𝐴 ∈ ℝ ) ∧ +∞ <ℝ 𝐴 ) ∨ ( +∞ = -∞ ∧ 𝐴 = +∞ ) ) ∨ ( ( +∞ ∈ ℝ ∧ 𝐴 = +∞ ) ∨ ( +∞ = -∞ ∧ 𝐴 ∈ ℝ ) ) )
13 pnfxr ⊢ +∞ ∈ ℝ*
14 ltxr ⊢ ( ( +∞ ∈ ℝ* ∧ 𝐴 ∈ ℝ* ) → ( +∞ < 𝐴 ↔ ( ( ( ( +∞ ∈ ℝ ∧ 𝐴 ∈ ℝ ) ∧ +∞ <ℝ 𝐴 ) ∨ ( +∞ = -∞ ∧ 𝐴 = +∞ ) ) ∨ ( ( +∞ ∈ ℝ ∧ 𝐴 = +∞ ) ∨ ( +∞ = -∞ ∧ 𝐴 ∈ ℝ ) ) ) ) )
15 13 14 mpan ⊢ ( 𝐴 ∈ ℝ* → ( +∞ < 𝐴 ↔ ( ( ( ( +∞ ∈ ℝ ∧ 𝐴 ∈ ℝ ) ∧ +∞ <ℝ 𝐴 ) ∨ ( +∞ = -∞ ∧ 𝐴 = +∞ ) ) ∨ ( ( +∞ ∈ ℝ ∧ 𝐴 = +∞ ) ∨ ( +∞ = -∞ ∧ 𝐴 ∈ ℝ ) ) ) ) )
16 12 15 mtbiri ⊢ ( 𝐴 ∈ ℝ* → ¬ +∞ < 𝐴 )