ubioc1 - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > ubioc1		GIF version

Theorem ubioc1 10066

Description: The upper bound belongs to an open-below, closed-above interval. See ubicc2 10122. (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 1001	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) → 𝐵 ∈ ℝ^*)
2		simp3 1002	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) → 𝐴 < 𝐵)
3		xrleid 9937	. . 3 ⊢ (𝐵 ∈ ℝ^* → 𝐵 ≤ 𝐵)
4	3	3ad2ant2 1022	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) → 𝐵 ≤ 𝐵)
5		elioc1 10059	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐵 ∈ (𝐴(,]𝐵) ↔ (𝐵 ∈ ℝ^ ∧ 𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐵)))
6	5	3adant3 1020	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) → (𝐵 ∈ (𝐴(,]𝐵) ↔ (𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵 ∧ 𝐵 ≤ 𝐵)))
7	1, 2, 4, 6	mpbir3and 1183	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) → 𝐵 ∈ (𝐴(,]𝐵))

Colors of variables: wff set class

Syntax hints: → wi 4 ↔ wb 105 ∧ w3a 981 ∈ wcel 2177 class class class wbr 4050 (class class class)co 5956 ℝ^*cxr 8121 < clt 8122 ≤ cle 8123 (,]cioc 10026

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 711 ax-5 1471 ax-7 1472 ax-gen 1473 ax-ie1 1517 ax-ie2 1518 ax-8 1528 ax-10 1529 ax-11 1530 ax-i12 1531 ax-bndl 1533 ax-4 1534 ax-17 1550 ax-i9 1554 ax-ial 1558 ax-i5r 1559 ax-13 2179 ax-14 2180 ax-ext 2188 ax-sep 4169 ax-pow 4225 ax-pr 4260 ax-un 4487 ax-setind 4592 ax-cnex 8031 ax-resscn 8032 ax-pre-ltirr 8052

This theorem depends on definitions: df-bi 117 df-3or 982 df-3an 983 df-tru 1376 df-fal 1379 df-nf 1485 df-sb 1787 df-eu 2058 df-mo 2059 df-clab 2193 df-cleq 2199 df-clel 2202 df-nfc 2338 df-ne 2378 df-nel 2473 df-ral 2490 df-rex 2491 df-rab 2494 df-v 2775 df-sbc 3003 df-dif 3172 df-un 3174 df-in 3176 df-ss 3183 df-pw 3622 df-sn 3643 df-pr 3644 df-op 3646 df-uni 3856 df-br 4051 df-opab 4113 df-id 4347 df-xp 4688 df-rel 4689 df-cnv 4690 df-co 4691 df-dm 4692 df-iota 5240 df-fun 5281 df-fv 5287 df-ov 5959 df-oprab 5960 df-mpo 5961 df-pnf 8124 df-mnf 8125 df-xr 8126 df-ltxr 8127 df-le 8128 df-ioc 10030

This theorem is referenced by: (None)

W3C validator