ubioo - Metamath Proof Explorer

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Theorem ubioo 13410

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 13407	. . . 4 ⊢ (𝐵 ∈ (𝐴(,)𝐵) ↔ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐵)))
2	1	simprbi 495	. . 3 ⊢ (𝐵 ∈ (𝐴(,)𝐵) → (𝐴 < 𝐵 ∧ 𝐵 < 𝐵))
3	2	simprd 494	. 2 ⊢ (𝐵 ∈ (𝐴(,)𝐵) → 𝐵 < 𝐵)
4	1	simplbi 496	. . . 4 ⊢ (𝐵 ∈ (𝐴(,)𝐵) → (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*))
5	4	simp2d 1140	. . 3 ⊢ (𝐵 ∈ (𝐴(,)𝐵) → 𝐵 ∈ ℝ^*)
6		xrltnr 13153	. . 3 ⊢ (𝐵 ∈ ℝ^* → ¬ 𝐵 < 𝐵)
7	5, 6	syl 17	. 2 ⊢ (𝐵 ∈ (𝐴(,)𝐵) → ¬ 𝐵 < 𝐵)
8	3, 7	pm2.65i 193	1 ⊢ ¬ 𝐵 ∈ (𝐴(,)𝐵)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 ∧ wa 394 ∧ w3a 1084 ∈ wcel 2099 class class class wbr 5153 (class class class)co 7424 ℝ^*cxr 11297 < clt 11298 (,)cioo 13378

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-resscn 11215 ax-pre-lttri 11232 ax-pre-lttrn 11233

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-id 5580 df-po 5594 df-so 5595 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-ov 7427 df-oprab 7428 df-mpo 7429 df-1st 8003 df-2nd 8004 df-er 8734 df-en 8975 df-dom 8976 df-sdom 8977 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-ioo 13382

This theorem is referenced by: lhop 26040 iooinlbub 45119 lptioo2 45252 volioc 45593 fourierdlem60 45787

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