ubioc1 - Metamath Proof Explorer

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Theorem ubioc1 13352

Description: The upper bound belongs to an open-below, closed-above interval. See ubicc2 13418. (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 1138	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) → 𝐵 ∈ ℝ^*)
2		simp3 1139	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) → 𝐴 < 𝐵)
3		xrleid 13102	. . 3 ⊢ (𝐵 ∈ ℝ^* → 𝐵 ≤ 𝐵)
4	3	3ad2ant2 1135	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) → 𝐵 ≤ 𝐵)
5		elioc1 13340	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐵 ∈ (𝐴(,]𝐵) ↔ (𝐵 ∈ ℝ^ ∧ 𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐵)))
6	5	3adant3 1133	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) → (𝐵 ∈ (𝐴(,]𝐵) ↔ (𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐵)))
7	1, 2, 4, 6	mpbir3and 1344	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) → 𝐵 ∈ (𝐴(,]𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ w3a 1087 ∈ wcel 2114 class class class wbr 5085 (class class class)co 7367 ℝ^*cxr 11178 < clt 11179 ≤ cle 11180 (,]cioc 13299

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-cnex 11094 ax-resscn 11095 ax-pre-lttri 11112 ax-pre-lttrn 11113

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4851 df-br 5086 df-opab 5148 df-mpt 5167 df-id 5526 df-po 5539 df-so 5540 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-ov 7370 df-oprab 7371 df-mpo 7372 df-er 8643 df-en 8894 df-dom 8895 df-sdom 8896 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-ioc 13303

This theorem is referenced by: xrlimcnp 26932 pnfneige0 34095 lmxrge0 34096 bj-minftyccb 37539 dvasin 38025 dvacos 38026 readvrec 42794 limcresiooub 46070 xlimpnfvlem1 46264 fourierdlem79 46613 fourierdlem93 46627 fouriersw 46659

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