ubioc1 - Metamath Proof Explorer

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Theorem ubioc1 13061

Description: The upper bound belongs to an open-below, closed-above interval. See ubicc2 13126. (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 1135	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) → 𝐵 ∈ ℝ^*)
2		simp3 1136	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) → 𝐴 < 𝐵)
3		xrleid 12814	. . 3 ⊢ (𝐵 ∈ ℝ^* → 𝐵 ≤ 𝐵)
4	3	3ad2ant2 1132	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) → 𝐵 ≤ 𝐵)
5		elioc1 13050	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐵 ∈ (𝐴(,]𝐵) ↔ (𝐵 ∈ ℝ^ ∧ 𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐵)))
6	5	3adant3 1130	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) → (𝐵 ∈ (𝐴(,]𝐵) ↔ (𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐵)))
7	1, 2, 4, 6	mpbir3and 1340	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) → 𝐵 ∈ (𝐴(,]𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ w3a 1085 ∈ wcel 2108 class class class wbr 5070 (class class class)co 7255 ℝ^*cxr 10939 < clt 10940 ≤ cle 10941 (,]cioc 13009

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-pre-lttri 10876 ax-pre-lttrn 10877

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-op 4565 df-uni 4837 df-br 5071 df-opab 5133 df-mpt 5154 df-id 5480 df-po 5494 df-so 5495 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-ov 7258 df-oprab 7259 df-mpo 7260 df-er 8456 df-en 8692 df-dom 8693 df-sdom 8694 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-ioc 13013

This theorem is referenced by: xrlimcnp 26023 pnfneige0 31803 lmxrge0 31804 bj-minftyccb 35323 dvasin 35788 dvacos 35789 limcresiooub 43073 xlimpnfvlem1 43267 fourierdlem79 43616 fourierdlem93 43630 fouriersw 43662

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