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Metamath Proof Explorer

Theorem mnfltpnf

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

Ref Expression
Assertion mnfltpnf -∞ < +∞

Proof

Step Hyp Ref Expression
1 eqid ⊢ -∞ = -∞
2 eqid ⊢ +∞ = +∞
3 olc ⊢ ( ( -∞ = -∞ ∧ +∞ = +∞ ) → ( ( ( -∞ ∈ ℝ ∧ +∞ ∈ ℝ ) ∧ -∞ <ℝ +∞ ) ∨ ( -∞ = -∞ ∧ +∞ = +∞ ) ) )
4 1 2 3 mp2an ⊢ ( ( ( -∞ ∈ ℝ ∧ +∞ ∈ ℝ ) ∧ -∞ <ℝ +∞ ) ∨ ( -∞ = -∞ ∧ +∞ = +∞ ) )
5 4 orci ⊢ ( ( ( ( -∞ ∈ ℝ ∧ +∞ ∈ ℝ ) ∧ -∞ <ℝ +∞ ) ∨ ( -∞ = -∞ ∧ +∞ = +∞ ) ) ∨ ( ( -∞ ∈ ℝ ∧ +∞ = +∞ ) ∨ ( -∞ = -∞ ∧ +∞ ∈ ℝ ) ) )
6 mnfxr ⊢ -∞ ∈ ℝ*
7 pnfxr ⊢ +∞ ∈ ℝ*
8 ltxr ⊢ ( ( -∞ ∈ ℝ* ∧ +∞ ∈ ℝ* ) → ( -∞ < +∞ ↔ ( ( ( ( -∞ ∈ ℝ ∧ +∞ ∈ ℝ ) ∧ -∞ <ℝ +∞ ) ∨ ( -∞ = -∞ ∧ +∞ = +∞ ) ) ∨ ( ( -∞ ∈ ℝ ∧ +∞ = +∞ ) ∨ ( -∞ = -∞ ∧ +∞ ∈ ℝ ) ) ) ) )
9 6 7 8 mp2an ⊢ ( -∞ < +∞ ↔ ( ( ( ( -∞ ∈ ℝ ∧ +∞ ∈ ℝ ) ∧ -∞ <ℝ +∞ ) ∨ ( -∞ = -∞ ∧ +∞ = +∞ ) ) ∨ ( ( -∞ ∈ ℝ ∧ +∞ = +∞ ) ∨ ( -∞ = -∞ ∧ +∞ ∈ ℝ ) ) ) )
10 5 9 mpbir ⊢ -∞ < +∞