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Theorem elico2 13428

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 11298	. . 3 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ^*)
2		elico1 13407	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐶 ∈ (𝐴[,)𝐵) ↔ (𝐶 ∈ ℝ^ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵)))
3	1, 2	sylan 578	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^) → (𝐶 ∈ (𝐴[,)𝐵) ↔ (𝐶 ∈ ℝ^ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵)))
4		mnfxr 11309	. . . . . . . 8 ⊢ -∞ ∈ ℝ^*
5	4	a1i 11	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵)) → -∞ ∈ ℝ^*)
6	1	ad2antrr 724	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵)) → 𝐴 ∈ ℝ^*)
7		simpr1 1191	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵)) → 𝐶 ∈ ℝ^*)
8		mnflt 13143	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ → -∞ < 𝐴)
9	8	ad2antrr 724	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵)) → -∞ < 𝐴)
10		simpr2 1192	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵)) → 𝐴 ≤ 𝐶)
11	5, 6, 7, 9, 10	xrltletrd 13180	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵)) → -∞ < 𝐶)
12		simplr 767	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵)) → 𝐵 ∈ ℝ^*)
13		pnfxr 11306	. . . . . . . 8 ⊢ +∞ ∈ ℝ^*
14	13	a1i 11	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵)) → +∞ ∈ ℝ^*)
15		simpr3 1193	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵)) → 𝐶 < 𝐵)
16		pnfge 13150	. . . . . . . 8 ⊢ (𝐵 ∈ ℝ^* → 𝐵 ≤ +∞)
17	16	ad2antlr 725	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵)) → 𝐵 ≤ +∞)
18	7, 12, 14, 15, 17	xrltletrd 13180	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵)) → 𝐶 < +∞)
19		xrrebnd 13187	. . . . . . 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 711	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵)) → 𝐶 ∈ ℝ)
22	21, 10, 15	3jca 1125	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵)) → (𝐶 ∈ ℝ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵))
23	22	ex 411	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^) → ((𝐶 ∈ ℝ^ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵) → (𝐶 ∈ ℝ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵)))
24		rexr 11298	. . . 4 ⊢ (𝐶 ∈ ℝ → 𝐶 ∈ ℝ^*)
25	24	3anim1i 1149	. . 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 394 ∧ w3a 1084 ∈ wcel 2098 class class class wbr 5152 (class class class)co 7426 ℝcr 11145 +∞cpnf 11283 -∞cmnf 11284 ℝ^*cxr 11285 < clt 11286 ≤ cle 11287 [,)cico 13366

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-pre-lttri 11220 ax-pre-lttrn 11221

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-po 5594 df-so 5595 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-ov 7429 df-oprab 7430 df-mpo 7431 df-er 8731 df-en 8971 df-dom 8972 df-sdom 8973 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-ico 13370

This theorem is referenced by: icossre 13445 elicopnf 13462 icoshft 13490 modelico 13886 muladdmodid 13916 icodiamlt 15422 fprodge0 15977 fprodge1 15979 rge0srg 21378 metustexhalf 24485 cnbl0 24710 icoopnst 24883 iocopnst 24884 icopnfcnv 24887 icopnfhmeo 24888 iccpnfcnv 24889 psercnlem2 26381 psercnlem1 26382 psercn 26383 abelth 26398 cosq34lt1 26481 tanord1 26491 tanord 26492 efopnlem1 26610 logtayl 26614 rlimcnp 26917 rlimcnp2 26918 dchrvmasumlem2 27451 dchrvmasumiflem1 27454 pntlemb 27550 pnt 27567 ubico 32564 xrge0slmod 33084 voliune 33881 volfiniune 33882 dya2icoseg 33930 sibfinima 33992 relowlpssretop 36876 tan2h 37118 itg2addnclem2 37178 binomcxplemdvbinom 43821 binomcxplemcvg 43822 binomcxplemnotnn0 43824 limciccioolb 45038 fourierdlem32 45556 fourierdlem43 45567 fourierdlem63 45586 fourierdlem79 45602 fouriersw 45648 expnegico01 47664 dignnld 47754 eenglngeehlnmlem1 47888 i0oii 48016 sepfsepc 48024

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