xrnmnfpnf - Mathbox for Glauco Siliprandi

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Theorem xrnmnfpnf 46245

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 13259	. . 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 1569 ∈ wcel 2146 ≠ wne 2959 ℝcr 11212 +∞cpnf 11353 -∞cmnf 11354 ℝ^*cxr 11355

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1944 ax-6 2001 ax-7 2042 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2196 ax-12 2215 ax-ext 2735 ax-sep 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395 ax-un 7761 ax-cnex 11269 ax-resscn 11270

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2101 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4839 df-uni 4871 df-br 5107 df-opab 5171 df-mpt 5190 df-id 5550 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-iota 6501 df-fun 6549 df-fn 6550 df-f 6551 df-f1 6552 df-fo 6553 df-f1o 6554 df-fv 6555 df-er 8731 df-en 8988 df-dom 8989 df-sdom 8990 df-pnf 11358 df-mnf 11359 df-xr 11360

This theorem is used by: infxr 46523 dfxlim2v 47002 xlimliminflimsup 47017

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