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Theorem lbico1 13455

Description: The lower bound belongs to a closed-below, open-above interval. See lbicc2 13519. (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 13204	. . 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 13443	. . 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 2145 class class class wbr 5107 (class class class)co 7416 ℝ^*cxr 11269 < clt 11270 ≤ cle 11271 [,)cico 13402

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2734 ax-sep 5255 ax-nul 5267 ax-pow 5334 ax-pr 5402 ax-un 7739 ax-cnex 11183 ax-resscn 11184 ax-pre-lttri 11201 ax-pre-lttrn 11202

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rab 3415 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-nul 4283 df-if 4486 df-pw 4562 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4871 df-br 5108 df-opab 5172 df-mpt 5191 df-id 5554 df-po 5567 df-so 5568 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-ov 7419 df-oprab 7420 df-mpo 7421 df-er 8699 df-en 8956 df-dom 8957 df-sdom 8958 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-ico 13406

This theorem is used by: icopnfsup 13928 metustid 24781 ioombl 25794 dchrvmasumlem2 27732 pntleme 27842 sxbrsigalem0 34769 icoreunrn 38100 dvasin 38440 dvacos 38441 limcresioolb 46458 xlimmnfvlem1 46647 fourierdlem93 47014

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