ubioo - Metamath Proof Explorer

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Theorem ubioo 13305

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 13302	. . . 4 ⊢ (𝐵 ∈ (𝐴(,)𝐵) ↔ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 < 𝐵 ∧ 𝐵 < 𝐵)))
2	1	simprbi 497	. . 3 ⊢ (𝐵 ∈ (𝐴(,)𝐵) → (𝐴 < 𝐵 ∧ 𝐵 < 𝐵))
3	2	simprd 495	. 2 ⊢ (𝐵 ∈ (𝐴(,)𝐵) → 𝐵 < 𝐵)
4	1	simplbi 496	. . . 4 ⊢ (𝐵 ∈ (𝐴(,)𝐵) → (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*))
5	4	simp2d 1144	. . 3 ⊢ (𝐵 ∈ (𝐴(,)𝐵) → 𝐵 ∈ ℝ^*)
6		xrltnr 13045	. . 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 1087 ∈ wcel 2114 class class class wbr 5100 (class class class)co 7368 ℝ^*cxr 11177 < clt 11178 (,)cioo 13273

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 2709 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 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 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-iun 4950 df-br 5101 df-opab 5163 df-mpt 5182 df-id 5527 df-po 5540 df-so 5541 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-ov 7371 df-oprab 7372 df-mpo 7373 df-1st 7943 df-2nd 7944 df-er 8645 df-en 8896 df-dom 8897 df-sdom 8898 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-ioo 13277

This theorem is referenced by: lhop 25989 iooinlbub 45855 lptioo2 45985 volioc 46324 fourierdlem60 46518

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