ubioo - Metamath Proof Explorer

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Theorem ubioo 13293

Description: An open interval does not contain its right endpoint. (Contributed by Mario Carneiro, 29-Dec-2016.)

**Assertion**
Ref	Expression
ubioo	⊢ ¬ 𝐵 ∈ (𝐴(,)𝐵)

**Proof of Theorem ubioo**
Step	Hyp	Ref	Expression
1		elioo3g 13290	. . . 4 ⊢ (𝐵 ∈ (𝐴(,)𝐵) ↔ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐵)))
2	1	simprbi 496	. . 3 ⊢ (𝐵 ∈ (𝐴(,)𝐵) → (𝐴 < 𝐵 ∧ 𝐵 < 𝐵))
3	2	simprd 495	. 2 ⊢ (𝐵 ∈ (𝐴(,)𝐵) → 𝐵 < 𝐵)
4	1	simplbi 497	. . . 4 ⊢ (𝐵 ∈ (𝐴(,)𝐵) → (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*))
5	4	simp2d 1143	. . 3 ⊢ (𝐵 ∈ (𝐴(,)𝐵) → 𝐵 ∈ ℝ^*)
6		xrltnr 13033	. . 3 ⊢ (𝐵 ∈ ℝ^* → ¬ 𝐵 < 𝐵)
7	5, 6	syl 17	. 2 ⊢ (𝐵 ∈ (𝐴(,)𝐵) → ¬ 𝐵 < 𝐵)
8	3, 7	pm2.65i 194	1 ⊢ ¬ 𝐵 ∈ (𝐴(,)𝐵)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 ∧ wa 395 ∧ w3a 1086 ∈ wcel 2113 class class class wbr 5098 (class class class)co 7358 ℝ^*cxr 11165 < clt 11166 (,)cioo 13261

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 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-cnex 11082 ax-resscn 11083 ax-pre-lttri 11100 ax-pre-lttrn 11101

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 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 3061 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-op 4587 df-uni 4864 df-iun 4948 df-br 5099 df-opab 5161 df-mpt 5180 df-id 5519 df-po 5532 df-so 5533 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-ov 7361 df-oprab 7362 df-mpo 7363 df-1st 7933 df-2nd 7934 df-er 8635 df-en 8884 df-dom 8885 df-sdom 8886 df-pnf 11168 df-mnf 11169 df-xr 11170 df-ltxr 11171 df-ioo 13265

This theorem is referenced by: lhop 25977 iooinlbub 45743 lptioo2 45873 volioc 46212 fourierdlem60 46406

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