xrre2 - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > xrre2		GIF version

Theorem xrre2 10202

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 10173	. . . . . . 7 ⊢ (𝐴 ∈ ℝ^* → -∞ ≤ 𝐴)
2	1	adantr 276	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → -∞ ≤ 𝐴)
3		mnfxr 8372	. . . . . . 7 ⊢ -∞ ∈ ℝ^*
4		xrlelttr 10187	. . . . . . 7 ⊢ ((-∞ ∈ ℝ^* ∧ 𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((-∞ ≤ 𝐴 ∧ 𝐴 < 𝐵) → -∞ < 𝐵))
5	3, 4	mp3an1 1365	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((-∞ ≤ 𝐴 ∧ 𝐴 < 𝐵) → -∞ < 𝐵))
6	2, 5	mpand 433	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 < 𝐵 → -∞ < 𝐵))
7	6	3adant3 1048	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐴 < 𝐵 → -∞ < 𝐵))
8		pnfge 10170	. . . . . . 7 ⊢ (𝐶 ∈ ℝ^* → 𝐶 ≤ +∞)
9	8	adantl 277	. . . . . 6 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → 𝐶 ≤ +∞)
10		pnfxr 8368	. . . . . . 7 ⊢ +∞ ∈ ℝ^*
11		xrltletr 10188	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → ((𝐵 < 𝐶 ∧ 𝐶 ≤ +∞) → 𝐵 < +∞))
12	10, 11	mp3an3 1367	. . . . . 6 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → ((𝐵 < 𝐶 ∧ 𝐶 ≤ +∞) → 𝐵 < +∞))
13	9, 12	mpan2d 432	. . . . 5 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐵 < 𝐶 → 𝐵 < +∞))
14	13	3adant1 1046	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐵 < 𝐶 → 𝐵 < +∞))
15	7, 14	anim12d 335	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → (-∞ < 𝐵 ∧ 𝐵 < +∞)))
16		xrrebnd 10200	. . . 4 ⊢ (𝐵 ∈ ℝ^* → (𝐵 ∈ ℝ ↔ (-∞ < 𝐵 ∧ 𝐵 < +∞)))
17	16	3ad2ant2 1050	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐵 ∈ ℝ ↔ (-∞ < 𝐵 ∧ 𝐵 < +∞)))
18	15, 17	sylibrd 169	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐵 ∈ ℝ))
19	18	imp 124	1 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → 𝐵 ∈ ℝ)

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∧ w3a 1009 ∈ wcel 2209 class class class wbr 4125 ℝcr 8168 +∞cpnf 8347 -∞cmnf 8348 ℝ^*cxr 8349 < clt 8350 ≤ cle 8351

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-sep 4244 ax-pow 4306 ax-pr 4341 ax-un 4573 ax-setind 4679 ax-cnex 8260 ax-resscn 8261 ax-pre-ltirr 8281 ax-pre-ltwlin 8282 ax-pre-lttrn 8283

This theorem depends on definitions: df-bi 117 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-rab 2537 df-v 2823 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-pw 3687 df-sn 3711 df-pr 3712 df-op 3714 df-uni 3931 df-br 4126 df-opab 4188 df-po 4436 df-iso 4437 df-xp 4775 df-cnv 4777 df-pnf 8352 df-mnf 8353 df-xr 8354 df-ltxr 8355 df-le 8356

This theorem is referenced by: elioore 10293 tgioo 15578

W3C validator