xrred - Mathbox for Glauco Siliprandi

	Mathbox for Glauco Siliprandi		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > xrred		Structured version Visualization version GIF version

Theorem xrred 46121

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

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

**Assertion**
Ref	Expression
xrred	⊢ (𝜑 → 𝐴 ∈ ℝ)

**Proof of Theorem xrred**
Step	Hyp	Ref	Expression
1		xrred.1	. . . 4 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
2		xrred.2	. . . 4 ⊢ (𝜑 → 𝐴 ≠ -∞)
3	1, 2	jca 521	. . 3 ⊢ (𝜑 → (𝐴 ∈ ℝ^* ∧ 𝐴 ≠ -∞))
4		xrnemnf 13160	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ≠ -∞) ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞))
5	3, 4	sylib 221	. 2 ⊢ (𝜑 → (𝐴 ∈ ℝ ∨ 𝐴 = +∞))
6		xrred.3	. . 3 ⊢ (𝜑 → 𝐴 ≠ +∞)
7	6	neneqd 2966	. 2 ⊢ (𝜑 → ¬ 𝐴 = +∞)
8		pm2.53 865	. . 3 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞) → (¬ 𝐴 ∈ ℝ → 𝐴 = +∞))
9	8	con1d 146	. 2 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞) → (¬ 𝐴 = +∞ → 𝐴 ∈ ℝ))
10	5, 7, 9	sylc 66	1 ⊢ (𝜑 → 𝐴 ∈ ℝ)

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 2961 ℝcr 11117 +∞cpnf 11258 -∞cmnf 11259 ℝ^*cxr 11260

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 2738 ax-sep 5262 ax-nul 5274 ax-pow 5341 ax-pr 5409 ax-un 7745 ax-cnex 11174 ax-resscn 11175

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 2570 df-eu 2600 df-clab 2745 df-cleq 2758 df-clel 2841 df-nfc 2915 df-ne 2962 df-nel 3068 df-ral 3083 df-rex 3093 df-rab 3420 df-v 3460 df-sbc 3748 df-csb 3857 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4290 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-op 4601 df-uni 4878 df-br 5115 df-opab 5179 df-mpt 5198 df-id 5561 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-iota 6499 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-er 8703 df-en 8953 df-dom 8954 df-sdom 8955 df-pnf 11263 df-mnf 11264 df-xr 11265

This theorem is used by: infxr 46123 infleinflem2 46127 xrralrecnnge 46146 xrre4 46166 supminfxr2 46224 xrpnf 46240 climxrrelem 46504 climxrre 46505 liminflimsupxrre 46572 ioorrnopnxrlem 47061 pimiooltgt 47465 smfpimltxr 47502 smfpimgtxr 47535

W3C validator