xrnmnfpnf - Mathbox for Glauco Siliprandi

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Theorem xrnmnfpnf 45880

Description: An extended real that is neither real nor minus infinity, is plus infinity. (Contributed by Glauco Siliprandi, 3-Mar-2021.)

**Hypotheses**
Ref	Expression
xrnmnfpnf.1	⊢ (𝜑 → 𝐴 ∈ ℝ^*)
xrnmnfpnf.2	⊢ (𝜑 → ¬ 𝐴 ∈ ℝ)
xrnmnfpnf.3	⊢ (𝜑 → 𝐴 ≠ -∞)

**Assertion**
Ref	Expression
xrnmnfpnf	⊢ (𝜑 → 𝐴 = +∞)

**Proof of Theorem xrnmnfpnf**
Step	Hyp	Ref	Expression
1		xrnmnfpnf.1	. . . 4 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
2		xrnmnfpnf.3	. . . 4 ⊢ (𝜑 → 𝐴 ≠ -∞)
3	1, 2	jca 521	. . 3 ⊢ (𝜑 → (𝐴 ∈ ℝ^* ∧ 𝐴 ≠ -∞))
4		xrnemnf 13162	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ≠ -∞) ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞))
5	3, 4	sylib 221	. 2 ⊢ (𝜑 → (𝐴 ∈ ℝ ∨ 𝐴 = +∞))
6		xrnmnfpnf.2	. 2 ⊢ (𝜑 → ¬ 𝐴 ∈ ℝ)
7		pm2.53 865	. 2 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞) → (¬ 𝐴 ∈ ℝ → 𝐴 = +∞))
8	5, 6, 7	sylc 66	1 ⊢ (𝜑 → 𝐴 = +∞)

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 401 ∨ wo 861 = wceq 1570 ∈ wcel 2146 ≠ wne 2960 ℝcr 11118 +∞cpnf 11259 -∞cmnf 11260 ℝ^*cxr 11261

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7742 ax-cnex 11175 ax-resscn 11176

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4875 df-br 5112 df-opab 5176 df-mpt 5195 df-id 5558 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-er 8700 df-en 8950 df-dom 8951 df-sdom 8952 df-pnf 11264 df-mnf 11265 df-xr 11266

This theorem is used by: infxr 46159 dfxlim2v 46638 xlimliminflimsup 46653

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