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Theorem elico2 12108

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 9964	. . 3 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ^*)
2		elico1 12089	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐶 ∈ (𝐴[,)𝐵) ↔ (𝐶 ∈ ℝ^ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵)))
3	1, 2	sylan 487	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^) → (𝐶 ∈ (𝐴[,)𝐵) ↔ (𝐶 ∈ ℝ^ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵)))
4		mnfxr 9975	. . . . . . . 8 ⊢ -∞ ∈ ℝ^*
5	4	a1i 11	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵)) → -∞ ∈ ℝ^*)
6	1	ad2antrr 758	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵)) → 𝐴 ∈ ℝ^*)
7		simpr1 1060	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵)) → 𝐶 ∈ ℝ^*)
8		mnflt 11833	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ → -∞ < 𝐴)
9	8	ad2antrr 758	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵)) → -∞ < 𝐴)
10		simpr2 1061	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵)) → 𝐴 ≤ 𝐶)
11	5, 6, 7, 9, 10	xrltletrd 11868	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵)) → -∞ < 𝐶)
12		simplr 788	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵)) → 𝐵 ∈ ℝ^*)
13		pnfxr 9971	. . . . . . . 8 ⊢ +∞ ∈ ℝ^*
14	13	a1i 11	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵)) → +∞ ∈ ℝ^*)
15		simpr3 1062	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵)) → 𝐶 < 𝐵)
16		pnfge 11840	. . . . . . . 8 ⊢ (𝐵 ∈ ℝ^* → 𝐵 ≤ +∞)
17	16	ad2antlr 759	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵)) → 𝐵 ≤ +∞)
18	7, 12, 14, 15, 17	xrltletrd 11868	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵)) → 𝐶 < +∞)
19		xrrebnd 11873	. . . . . . 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 959	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵)) → 𝐶 ∈ ℝ)
22	21, 10, 15	3jca 1235	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^) ∧ (𝐶 ∈ ℝ^ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵)) → (𝐶 ∈ ℝ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵))
23	22	ex 449	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^) → ((𝐶 ∈ ℝ^ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵) → (𝐶 ∈ ℝ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵)))
24		rexr 9964	. . . 4 ⊢ (𝐶 ∈ ℝ → 𝐶 ∈ ℝ^*)
25	24	3anim1i 1241	. . 3 ⊢ ((𝐶 ∈ ℝ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵) → (𝐶 ∈ ℝ^* ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵))
26	23, 25	impbid1 214	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^) → ((𝐶 ∈ ℝ^ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵) ↔ (𝐶 ∈ ℝ ∧ 𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵)))
27	3, 26	bitrd 267	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 195 ∧ wa 383 ∧ w3a 1031 ∈ wcel 1977 class class class wbr 4583 (class class class)co 6549 ℝcr 9814 +∞cpnf 9950 -∞cmnf 9951 ℝ^*cxr 9952 < clt 9953 ≤ cle 9954 [,)cico 12048

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-pre-lttri 9889 ax-pre-lttrn 9890

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-po 4959 df-so 4960 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-ico 12052

This theorem is referenced by: icossre 12125 elicopnf 12140 icoshft 12165 modelico 12542 muladdmodid 12572 icodiamlt 14022 fprodge0 14563 fprodge1 14565 rge0srg 19636 metustexhalf 22171 cnbl0 22387 icoopnst 22546 iocopnst 22547 icopnfcnv 22549 icopnfhmeo 22550 iccpnfcnv 22551 psercnlem2 23982 psercnlem1 23983 psercn 23984 abelth 23999 tanord1 24087 tanord 24088 efopnlem1 24202 logtayl 24206 rlimcnp 24492 rlimcnp2 24493 dchrvmasumlem2 24987 dchrvmasumiflem1 24990 pntlemb 25086 pnt 25103 ubico 28927 xrge0slmod 29175 voliune 29619 volfiniune 29620 dya2icoseg 29666 sibfinima 29728 relowlpssretop 32388 tan2h 32571 itg2addnclem2 32632 binomcxplemdvbinom 37574 binomcxplemcvg 37575 binomcxplemnotnn0 37577 limciccioolb 38688 fourierdlem32 39032 fourierdlem43 39043 fourierdlem63 39062 fourierdlem79 39078 fouriersw 39124 expnegico01 42102 dignnld 42195

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