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Theorem lbico1 13531

Description: The lower bound belongs to a closed-below, open-above interval. See lbicc2 13595. (Contributed by FL, 29-May-2014.)

**Assertion**
Ref	Expression
lbico1	⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) → 𝐴 ∈ (𝐴[,)𝐵))

**Proof of Theorem lbico1**
Step	Hyp	Ref	Expression
1		simp1 1154	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) → 𝐴 ∈ ℝ^*)
2		xrleid 13280	. . 3 ⊢ (𝐴 ∈ ℝ^* → 𝐴 ≤ 𝐴)
3	2	3ad2ant1 1151	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) → 𝐴 ≤ 𝐴)
4		simp3 1156	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) → 𝐴 < 𝐵)
5		elico1 13519	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐴 ∈ (𝐴[,)𝐵) ↔ (𝐴 ∈ ℝ^ ∧ 𝐴 ≤ 𝐴 ∧ 𝐴 < 𝐵)))
6	5	3adant3 1150	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) → (𝐴 ∈ (𝐴[,)𝐵) ↔ (𝐴 ∈ ℝ^* ∧ 𝐴 ≤ 𝐴 ∧ 𝐴 < 𝐵)))
7	1, 3, 4, 6	mpbir3and 1361	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) → 𝐴 ∈ (𝐴[,)𝐵))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ w3a 1103 ∈ wcel 2144 class class class wbr 5102 (class class class)co 7420 ℝ^*cxr 11342 < clt 11343 ≤ cle 11344 [,)cico 13478

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7751 ax-cnex 11256 ax-resscn 11257 ax-pre-lttri 11274 ax-pre-lttrn 11275

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4584 df-pr 4586 df-op 4590 df-uni 4867 df-br 5103 df-opab 5167 df-mpt 5186 df-id 5545 df-po 5558 df-so 5559 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-ov 7423 df-oprab 7424 df-mpo 7425 df-er 8717 df-en 8974 df-dom 8975 df-sdom 8976 df-pnf 11345 df-mnf 11346 df-xr 11347 df-ltxr 11348 df-le 11349 df-ico 13482

This theorem is used by: icopnfsup 14005 metustid 24873 ioombl 25886 dchrvmasumlem2 27825 pntleme 27935 sxbrsigalem0 34903 icoreunrn 38283 dvasin 38623 dvacos 38624 limcresioolb 46653 xlimmnfvlem1 46842 fourierdlem93 47209

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