Theorem xrnpnfmnf 46049

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 519	. . 3 ⊢ (𝜑 → (𝐴 ∈ ℝ* ∧ 𝐴 ≠ +∞))
4		xrnepnf 13121	. . 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		pm2.53 862	. 2 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = -∞) → (¬ 𝐴 ∈ ℝ → 𝐴 = -∞))
8	5, 6, 7	sylc 65	1 ⊢ (𝜑 → 𝐴 = -∞)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∨ wo 858   = wceq 1561   ∈ wcel 2143   ≠ wne 2958  ℝcr 11073  +∞cpnf 11214  -∞cmnf 11215  ℝ^*cxr 11216

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5247  ax-pow 5323  ax-un 7719  ax-cnex 11130  ax-resscn 11131

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1564  df-ex 1801  df-sb 2092  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-nel 3063  df-rab 3416  df-v 3457  df-un 3910  df-in 3912  df-ss 3922  df-pw 4558  df-sn 4584  df-pr 4586  df-uni 4867  df-pnf 11219  df-mnf 11220  df-xr 11221

This theorem is referenced by:  xlimliminflimsup  46437