lbico1 - Metamath Proof Explorer

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Theorem lbico1 13486

Description: The lower bound belongs to a closed-below, open-above interval. See lbicc2 13550. (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 13235	. . 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 13474	. . 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 5103 (class class class)co 7409 ℝ^*cxr 11299 < clt 11300 ≤ cle 11301 [,)cico 13433

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 2213 ax-ext 2732 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7735 ax-cnex 11213 ax-resscn 11214 ax-pre-lttri 11231 ax-pre-lttrn 11232

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 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 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-br 5104 df-opab 5168 df-mpt 5187 df-id 5543 df-po 5556 df-so 5557 df-xp 5654 df-rel 5655 df-cnv 5656 df-co 5657 df-dm 5658 df-rn 5659 df-res 5660 df-ima 5661 df-iota 6484 df-fun 6530 df-fn 6531 df-f 6532 df-f1 6533 df-fo 6534 df-f1o 6535 df-fv 6536 df-ov 7412 df-oprab 7413 df-mpo 7414 df-er 8696 df-en 8953 df-dom 8954 df-sdom 8955 df-pnf 11302 df-mnf 11303 df-xr 11304 df-ltxr 11305 df-le 11306 df-ico 13437

This theorem is used by: icopnfsup 13959 metustid 24820 ioombl 25833 dchrvmasumlem2 27774 pntleme 27884 sxbrsigalem0 34823 icoreunrn 38196 dvasin 38536 dvacos 38537 limcresioolb 46569 xlimmnfvlem1 46758 fourierdlem93 47125

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