ubioo - Metamath Proof Explorer

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Theorem ubioo 13362

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 13359	. . . 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 1141	. . 3 ⊢ (𝐵 ∈ (𝐴(,)𝐵) → 𝐵 ∈ ℝ^*)
6		xrltnr 13105	. . 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 1085 ∈ wcel 2104 class class class wbr 5149 (class class class)co 7413 ℝ^*cxr 11253 < clt 11254 (,)cioo 13330

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7729 ax-cnex 11170 ax-resscn 11171 ax-pre-lttri 11188 ax-pre-lttrn 11189

This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rab 3431 df-v 3474 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-po 5589 df-so 5590 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-ov 7416 df-oprab 7417 df-mpo 7418 df-1st 7979 df-2nd 7980 df-er 8707 df-en 8944 df-dom 8945 df-sdom 8946 df-pnf 11256 df-mnf 11257 df-xr 11258 df-ltxr 11259 df-ioo 13334

This theorem is referenced by: lhop 25767 iooinlbub 44514 lptioo2 44647 volioc 44988 fourierdlem60 45182

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