Theorem xrnpnfmnf 41758

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 514	. . 3 ⊢ (𝜑 → (𝐴 ∈ ℝ* ∧ 𝐴 ≠ +∞))
4		xrnepnf 12516	. . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ +∞) ↔ (𝐴 ∈ ℝ ∨ 𝐴 = -∞))
5	3, 4	sylib 220	. 2 ⊢ (𝜑 → (𝐴 ∈ ℝ ∨ 𝐴 = -∞))
6		xrnpnfmnf.2	. 2 ⊢ (𝜑 → ¬ 𝐴 ∈ ℝ)
7		pm2.53 847	. 2 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = -∞) → (¬ 𝐴 ∈ ℝ → 𝐴 = -∞))
8	5, 6, 7	sylc 65	1 ⊢ (𝜑 → 𝐴 = -∞)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398   ∨ wo 843   = wceq 1537   ∈ wcel 2114   ≠ wne 3018  ℝcr 10538  +∞cpnf 10674  -∞cmnf 10675  ℝ^*cxr 10676

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463  ax-cnex 10595  ax-resscn 10596

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-nel 3126  df-rab 3149  df-v 3498  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-pw 4543  df-sn 4570  df-pr 4572  df-uni 4841  df-pnf 10679  df-mnf 10680  df-xr 10681

This theorem is referenced by:  xlimliminflimsup  42150