xrre2 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > xrre2		Structured version Visualization version GIF version

Theorem xrre2 13184

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 13148	. . . . . . 7 ⊢ (𝐴 ∈ ℝ^* → -∞ ≤ 𝐴)
2	1	adantr 485	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → -∞ ≤ 𝐴)
3		mnfxr 11254	. . . . . . 7 ⊢ -∞ ∈ ℝ^*
4		xrlelttr 13169	. . . . . . 7 ⊢ ((-∞ ∈ ℝ^* ∧ 𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((-∞ ≤ 𝐴 ∧ 𝐴 < 𝐵) → -∞ < 𝐵))
5	3, 4	mp3an1 1472	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((-∞ ≤ 𝐴 ∧ 𝐴 < 𝐵) → -∞ < 𝐵))
6	2, 5	mpand 707	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 < 𝐵 → -∞ < 𝐵))
7	6	3adant3 1148	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐴 < 𝐵 → -∞ < 𝐵))
8		pnfge 13143	. . . . . . 7 ⊢ (𝐶 ∈ ℝ^* → 𝐶 ≤ +∞)
9	8	adantl 486	. . . . . 6 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → 𝐶 ≤ +∞)
10		pnfxr 11251	. . . . . . 7 ⊢ +∞ ∈ ℝ^*
11		xrltletr 13170	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → ((𝐵 < 𝐶 ∧ 𝐶 ≤ +∞) → 𝐵 < +∞))
12	10, 11	mp3an3 1474	. . . . . 6 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → ((𝐵 < 𝐶 ∧ 𝐶 ≤ +∞) → 𝐵 < +∞))
13	9, 12	mpan2d 706	. . . . 5 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐵 < 𝐶 → 𝐵 < +∞))
14	13	3adant1 1146	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐵 < 𝐶 → 𝐵 < +∞))
15	7, 14	anim12d 620	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → (-∞ < 𝐵 ∧ 𝐵 < +∞)))
16		xrrebnd 13182	. . . 4 ⊢ (𝐵 ∈ ℝ^* → (𝐵 ∈ ℝ ↔ (-∞ < 𝐵 ∧ 𝐵 < +∞)))
17	16	3ad2ant2 1150	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐵 ∈ ℝ ↔ (-∞ < 𝐵 ∧ 𝐵 < +∞)))
18	15, 17	sylibrd 262	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐵 ∈ ℝ))
19	18	imp 411	1 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐶)) → 𝐵 ∈ ℝ)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 400 ∧ w3a 1101 ∈ wcel 2145 class class class wbr 5104 ℝcr 11087 +∞cpnf 11228 -∞cmnf 11229 ℝ^*cxr 11230 < clt 11231 ≤ cle 11232

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1818 ax-4 1832 ax-5 1933 ax-6 1990 ax-7 2031 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2737 ax-sep 5250 ax-nul 5260 ax-pow 5326 ax-pr 5394 ax-un 7722 ax-cnex 11144 ax-resscn 11145 ax-pre-lttri 11162 ax-pre-lttrn 11163

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1566 df-fal 1576 df-ex 1803 df-nf 1807 df-sb 2094 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3065 df-ral 3080 df-rex 3090 df-rab 3418 df-v 3459 df-sbc 3748 df-csb 3856 df-dif 3910 df-un 3912 df-in 3914 df-ss 3924 df-nul 4289 df-if 4484 df-pw 4560 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4868 df-br 5105 df-opab 5167 df-mpt 5186 df-id 5546 df-po 5559 df-so 5560 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-iota 6481 df-fun 6527 df-fn 6528 df-f 6529 df-f1 6530 df-fo 6531 df-f1o 6532 df-fv 6533 df-er 8682 df-en 8932 df-dom 8933 df-sdom 8934 df-pnf 11233 df-mnf 11234 df-xr 11235 df-ltxr 11236 df-le 11237

This theorem is referenced by: elioore 13390 xrsdsreclblem 21520 pnfnei 23334 mnfnei 23335 tgioo 24910 ovolunnul 25616 icombl 25680 ioombl 25681 ioorcl2 25688 volivth 25723 dvferm2lem 26102 itg2gt0cn 38181 iccpartipre 48026

W3C validator