xrre2 - Intuitionistic Logic Explorer

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Theorem xrre2 9491

Description: An extended real between two others is real. (Contributed by NM, 6-Feb-2007.)

**Assertion**
Ref	Expression
xrre2	⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → 𝐵 ∈ ℝ)

**Proof of Theorem xrre2**
Step	Hyp	Ref	Expression
1		mnfle 9465	. . . . . . 7 ⊢ (𝐴 ∈ ℝ^* → -∞ ≤ 𝐴)
2	1	adantr 272	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → -∞ ≤ 𝐴)
3		mnfxr 7740	. . . . . . 7 ⊢ -∞ ∈ ℝ^*
4		xrlelttr 9476	. . . . . . 7 ⊢ ((-∞ ∈ ℝ^* ∧ 𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((-∞ ≤ 𝐴 ∧ 𝐴 < 𝐵) → -∞ < 𝐵))
5	3, 4	mp3an1 1283	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((-∞ ≤ 𝐴 ∧ 𝐴 < 𝐵) → -∞ < 𝐵))
6	2, 5	mpand 423	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 < 𝐵 → -∞ < 𝐵))
7	6	3adant3 982	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐴 < 𝐵 → -∞ < 𝐵))
8		pnfge 9462	. . . . . . 7 ⊢ (𝐶 ∈ ℝ^* → 𝐶 ≤ +∞)
9	8	adantl 273	. . . . . 6 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → 𝐶 ≤ +∞)
10		pnfxr 7736	. . . . . . 7 ⊢ +∞ ∈ ℝ^*
11		xrltletr 9477	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → ((𝐵 < 𝐶 ∧ 𝐶 ≤ +∞) → 𝐵 < +∞))
12	10, 11	mp3an3 1285	. . . . . 6 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → ((𝐵 < 𝐶 ∧ 𝐶 ≤ +∞) → 𝐵 < +∞))
13	9, 12	mpan2d 422	. . . . 5 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐵 < 𝐶 → 𝐵 < +∞))
14	13	3adant1 980	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐵 < 𝐶 → 𝐵 < +∞))
15	7, 14	anim12d 331	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → (-∞ < 𝐵 ∧ 𝐵 < +∞)))
16		xrrebnd 9489	. . . 4 ⊢ (𝐵 ∈ ℝ^* → (𝐵 ∈ ℝ ↔ (-∞ < 𝐵 ∧ 𝐵 < +∞)))
17	16	3ad2ant2 984	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐵 ∈ ℝ ↔ (-∞ < 𝐵 ∧ 𝐵 < +∞)))
18	15, 17	sylibrd 168	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐵 ∈ ℝ))
19	18	imp 123	1 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → 𝐵 ∈ ℝ)

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∧ w3a 943 ∈ wcel 1461 class class class wbr 3893 ℝcr 7540 +∞cpnf 7715 -∞cmnf 7716 ℝ^*cxr 7717 < clt 7718 ≤ cle 7719

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 586 ax-in2 587 ax-io 681 ax-5 1404 ax-7 1405 ax-gen 1406 ax-ie1 1450 ax-ie2 1451 ax-8 1463 ax-10 1464 ax-11 1465 ax-i12 1466 ax-bndl 1467 ax-4 1468 ax-13 1472 ax-14 1473 ax-17 1487 ax-i9 1491 ax-ial 1495 ax-i5r 1496 ax-ext 2095 ax-sep 4004 ax-pow 4056 ax-pr 4089 ax-un 4313 ax-setind 4410 ax-cnex 7630 ax-resscn 7631 ax-pre-ltirr 7651 ax-pre-ltwlin 7652 ax-pre-lttrn 7653

This theorem depends on definitions: df-bi 116 df-3or 944 df-3an 945 df-tru 1315 df-fal 1318 df-nf 1418 df-sb 1717 df-eu 1976 df-mo 1977 df-clab 2100 df-cleq 2106 df-clel 2109 df-nfc 2242 df-ne 2281 df-nel 2376 df-ral 2393 df-rex 2394 df-rab 2397 df-v 2657 df-dif 3037 df-un 3039 df-in 3041 df-ss 3048 df-pw 3476 df-sn 3497 df-pr 3498 df-op 3500 df-uni 3701 df-br 3894 df-opab 3948 df-po 4176 df-iso 4177 df-xp 4503 df-cnv 4505 df-pnf 7720 df-mnf 7721 df-xr 7722 df-ltxr 7723 df-le 7724

This theorem is referenced by: elioore 9582 tgioo 12526

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