Theorem xrnpnfmnf 46114

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 520	. . 3 ⊢ (𝜑 → (𝐴 ∈ ℝ* ∧ 𝐴 ≠ +∞))
4		xrnepnf 13143	. . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ +∞) ↔ (𝐴 ∈ ℝ ∨ 𝐴 = -∞))
5	3, 4	sylib 221	. 2 ⊢ (𝜑 → (𝐴 ∈ ℝ ∨ 𝐴 = -∞))
6		xrnpnfmnf.2	. 2 ⊢ (𝜑 → ¬ 𝐴 ∈ ℝ)
7		pm2.53 864	. 2 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = -∞) → (¬ 𝐴 ∈ ℝ → 𝐴 = -∞))
8	5, 6, 7	sylc 66	1 ⊢ (𝜑 → 𝐴 = -∞)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   ∨ wo 860   = wceq 1567   ∈ wcel 2149   ≠ wne 2964  ℝcr 11099  +∞cpnf 11240  -∞cmnf 11241  ℝ^*cxr 11242

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5261  ax-pow 5337  ax-un 7733  ax-cnex 11156  ax-resscn 11157

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ne 2965  df-nel 3071  df-rab 3424  df-v 3465  df-un 3918  df-in 3920  df-ss 3930  df-pw 4569  df-sn 4595  df-pr 4597  df-uni 4877  df-pnf 11245  df-mnf 11246  df-xr 11247

This theorem is referenced by:  xlimliminflimsup  46502