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Theorem elico2 13143

Description: Membership in a closed-below, open-above real interval. (Contributed by Paul Chapman, 21-Jan-2008.) (Revised by Mario Carneiro, 14-Jun-2014.)

**Assertion**
Ref	Expression
elico2	⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) → (𝐶 ∈ (𝐴[,)𝐵) ↔ (𝐶 ∈ ℝ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵)))

**Proof of Theorem elico2**
Step	Hyp	Ref	Expression
1		rexr 11021	. . 3 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ^*)
2		elico1 13122	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐶 ∈ (𝐴[,)𝐵) ↔ (𝐶 ∈ ℝ^ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵)))
3	1, 2	sylan 580	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^) → (𝐶 ∈ (𝐴[,)𝐵) ↔ (𝐶 ∈ ℝ^ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵)))
4		mnfxr 11032	. . . . . . . 8 ⊢ -∞ ∈ ℝ^*
5	4	a1i 11	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵)) → -∞ ∈ ℝ^*)
6	1	ad2antrr 723	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵)) → 𝐴 ∈ ℝ^*)
7		simpr1 1193	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵)) → 𝐶 ∈ ℝ^*)
8		mnflt 12859	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ → -∞ < 𝐴)
9	8	ad2antrr 723	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵)) → -∞ < 𝐴)
10		simpr2 1194	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵)) → 𝐴 ≤ 𝐶)
11	5, 6, 7, 9, 10	xrltletrd 12895	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵)) → -∞ < 𝐶)
12		simplr 766	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵)) → 𝐵 ∈ ℝ^*)
13		pnfxr 11029	. . . . . . . 8 ⊢ +∞ ∈ ℝ^*
14	13	a1i 11	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵)) → +∞ ∈ ℝ^*)
15		simpr3 1195	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵)) → 𝐶 < 𝐵)
16		pnfge 12866	. . . . . . . 8 ⊢ (𝐵 ∈ ℝ^* → 𝐵 ≤ +∞)
17	16	ad2antlr 724	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵)) → 𝐵 ≤ +∞)
18	7, 12, 14, 15, 17	xrltletrd 12895	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵)) → 𝐶 < +∞)
19		xrrebnd 12902	. . . . . . 7 ⊢ (𝐶 ∈ ℝ^* → (𝐶 ∈ ℝ ↔ (-∞ < 𝐶 ∧ 𝐶 < +∞)))
20	7, 19	syl 17	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵)) → (𝐶 ∈ ℝ ↔ (-∞ < 𝐶 ∧ 𝐶 < +∞)))
21	11, 18, 20	mpbir2and 710	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵)) → 𝐶 ∈ ℝ)
22	21, 10, 15	3jca 1127	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵)) → (𝐶 ∈ ℝ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵))
23	22	ex 413	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^) → ((𝐶 ∈ ℝ^ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵) → (𝐶 ∈ ℝ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵)))
24		rexr 11021	. . . 4 ⊢ (𝐶 ∈ ℝ → 𝐶 ∈ ℝ^*)
25	24	3anim1i 1151	. . 3 ⊢ ((𝐶 ∈ ℝ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵) → (𝐶 ∈ ℝ^* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵))
26	23, 25	impbid1 224	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^) → ((𝐶 ∈ ℝ^ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵) ↔ (𝐶 ∈ ℝ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵)))
27	3, 26	bitrd 278	1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) → (𝐶 ∈ (𝐴[,)𝐵) ↔ (𝐶 ∈ ℝ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1086 ∈ wcel 2106 class class class wbr 5074 (class class class)co 7275 ℝcr 10870 +∞cpnf 11006 -∞cmnf 11007 ℝ^*cxr 11008 < clt 11009 ≤ cle 11010 [,)cico 13081

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-pre-lttri 10945 ax-pre-lttrn 10946

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-po 5503 df-so 5504 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-ov 7278 df-oprab 7279 df-mpo 7280 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-ico 13085

This theorem is referenced by: icossre 13160 elicopnf 13177 icoshft 13205 modelico 13601 muladdmodid 13631 icodiamlt 15147 fprodge0 15703 fprodge1 15705 rge0srg 20669 metustexhalf 23712 cnbl0 23937 icoopnst 24102 iocopnst 24103 icopnfcnv 24105 icopnfhmeo 24106 iccpnfcnv 24107 psercnlem2 25583 psercnlem1 25584 psercn 25585 abelth 25600 cosq34lt1 25683 tanord1 25693 tanord 25694 efopnlem1 25811 logtayl 25815 rlimcnp 26115 rlimcnp2 26116 dchrvmasumlem2 26646 dchrvmasumiflem1 26649 pntlemb 26745 pnt 26762 ubico 31096 xrge0slmod 31548 voliune 32197 volfiniune 32198 dya2icoseg 32244 sibfinima 32306 relowlpssretop 35535 tan2h 35769 itg2addnclem2 35829 binomcxplemdvbinom 41971 binomcxplemcvg 41972 binomcxplemnotnn0 41974 limciccioolb 43162 fourierdlem32 43680 fourierdlem43 43691 fourierdlem63 43710 fourierdlem79 43726 fouriersw 43772 expnegico01 45859 dignnld 45949 eenglngeehlnmlem1 46083 i0oii 46213 sepfsepc 46221

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