Theorem xrnpnfmnf 45463

Description: An extended real that is neither real nor plus infinity, is minus infinity. (Contributed by Glauco Siliprandi, 5-Feb-2022.)

Hypotheses

Ref	Expression
xrnpnfmnf.1	⊢ (𝜑 → 𝐴 ∈ ℝ*)
xrnpnfmnf.2	⊢ (𝜑 → ¬ 𝐴 ∈ ℝ)
xrnpnfmnf.3	⊢ (𝜑 → 𝐴 ≠ +∞)

Assertion

Ref	Expression
xrnpnfmnf	⊢ (𝜑 → 𝐴 = -∞)

Proof of Theorem xrnpnfmnf

Step	Hyp	Ref	Expression
1		xrnpnfmnf.1	. . . 4 ⊢ (𝜑 → 𝐴 ∈ ℝ*)
2		xrnpnfmnf.3	. . . 4 ⊢ (𝜑 → 𝐴 ≠ +∞)
3	1, 2	jca 511	. . 3 ⊢ (𝜑 → (𝐴 ∈ ℝ* ∧ 𝐴 ≠ +∞))
4		xrnepnf 13054	. . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ +∞) ↔ (𝐴 ∈ ℝ ∨ 𝐴 = -∞))
5	3, 4	sylib 218	. 2 ⊢ (𝜑 → (𝐴 ∈ ℝ ∨ 𝐴 = -∞))
6		xrnpnfmnf.2	. 2 ⊢ (𝜑 → ¬ 𝐴 ∈ ℝ)
7		pm2.53 851	. 2 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = -∞) → (¬ 𝐴 ∈ ℝ → 𝐴 = -∞))
8	5, 6, 7	sylc 65	1 ⊢ (𝜑 → 𝐴 = -∞)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 847   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ℝcr 11043  +∞cpnf 11181  -∞cmnf 11182  ℝ^*cxr 11183

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5246  ax-pow 5315  ax-pr 5382  ax-un 7691  ax-cnex 11100  ax-resscn 11101

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-nel 3030  df-rab 3403  df-v 3446  df-un 3916  df-in 3918  df-ss 3928  df-pw 4561  df-sn 4586  df-pr 4588  df-uni 4868  df-pnf 11186  df-mnf 11187  df-xr 11188

This theorem is referenced by:  xlimliminflimsup  45853