ubioc1 - Metamath Proof Explorer

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Theorem ubioc1 13299

Description: The upper bound belongs to an open-below, closed-above interval. See ubicc2 13365. (Contributed by FL, 29-May-2014.)

**Assertion**
Ref	Expression
ubioc1	⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) → 𝐵 ∈ (𝐴(,]𝐵))

**Proof of Theorem ubioc1**
Step	Hyp	Ref	Expression
1		simp2 1137	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) → 𝐵 ∈ ℝ^*)
2		simp3 1138	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) → 𝐴 < 𝐵)
3		xrleid 13050	. . 3 ⊢ (𝐵 ∈ ℝ^* → 𝐵 ≤ 𝐵)
4	3	3ad2ant2 1134	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) → 𝐵 ≤ 𝐵)
5		elioc1 13287	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐵 ∈ (𝐴(,]𝐵) ↔ (𝐵 ∈ ℝ^ ∧ 𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐵)))
6	5	3adant3 1132	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) → (𝐵 ∈ (𝐴(,]𝐵) ↔ (𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐵)))
7	1, 2, 4, 6	mpbir3and 1343	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) → 𝐵 ∈ (𝐴(,]𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ w3a 1086 ∈ wcel 2111 class class class wbr 5091 (class class class)co 7346 ℝ^*cxr 11145 < clt 11146 ≤ cle 11147 (,]cioc 13246

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-sep 5234 ax-nul 5244 ax-pow 5303 ax-pr 5370 ax-un 7668 ax-cnex 11062 ax-resscn 11063 ax-pre-lttri 11080 ax-pre-lttrn 11081

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rab 3396 df-v 3438 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-nul 4284 df-if 4476 df-pw 4552 df-sn 4577 df-pr 4579 df-op 4583 df-uni 4860 df-br 5092 df-opab 5154 df-mpt 5173 df-id 5511 df-po 5524 df-so 5525 df-xp 5622 df-rel 5623 df-cnv 5624 df-co 5625 df-dm 5626 df-rn 5627 df-res 5628 df-ima 5629 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-ov 7349 df-oprab 7350 df-mpo 7351 df-er 8622 df-en 8870 df-dom 8871 df-sdom 8872 df-pnf 11148 df-mnf 11149 df-xr 11150 df-ltxr 11151 df-le 11152 df-ioc 13250

This theorem is referenced by: xrlimcnp 26906 pnfneige0 33962 lmxrge0 33963 bj-minftyccb 37265 dvasin 37750 dvacos 37751 readvrec 42401 limcresiooub 45686 xlimpnfvlem1 45880 fourierdlem79 46229 fourierdlem93 46243 fouriersw 46275

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