icoub - Mathbox for Glauco Siliprandi

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Theorem icoub 44226

Description: A left-closed, right-open interval does not contain its upper bound. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

**Assertion**
Ref	Expression
icoub	⊢ (𝐴 ∈ ℝ^* → ¬ 𝐵 ∈ (𝐴[,)𝐵))

**Proof of Theorem icoub**
Step	Hyp	Ref	Expression
1		simpl 484	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ (𝐴[,)𝐵)) → 𝐴 ∈ ℝ^*)
2		icossxr 13406	. . . . 5 ⊢ (𝐴[,)𝐵) ⊆ ℝ^*
3		id 22	. . . . 5 ⊢ (𝐵 ∈ (𝐴[,)𝐵) → 𝐵 ∈ (𝐴[,)𝐵))
4	2, 3	sselid 3980	. . . 4 ⊢ (𝐵 ∈ (𝐴[,)𝐵) → 𝐵 ∈ ℝ^*)
5	4	adantl 483	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ (𝐴[,)𝐵)) → 𝐵 ∈ ℝ^*)
6		simpr 486	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ (𝐴[,)𝐵)) → 𝐵 ∈ (𝐴[,)𝐵))
7		icoltub 44208	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐵 ∈ (𝐴[,)𝐵)) → 𝐵 < 𝐵)
8	1, 5, 6, 7	syl3anc 1372	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ (𝐴[,)𝐵)) → 𝐵 < 𝐵)
9		xrltnr 13096	. . . 4 ⊢ (𝐵 ∈ ℝ^* → ¬ 𝐵 < 𝐵)
10	4, 9	syl 17	. . 3 ⊢ (𝐵 ∈ (𝐴[,)𝐵) → ¬ 𝐵 < 𝐵)
11	10	adantl 483	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ (𝐴[,)𝐵)) → ¬ 𝐵 < 𝐵)
12	8, 11	pm2.65da 816	1 ⊢ (𝐴 ∈ ℝ^* → ¬ 𝐵 ∈ (𝐴[,)𝐵))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 397 ∈ wcel 2107 class class class wbr 5148 (class class class)co 7406 ℝ^*cxr 11244 < clt 11245 [,)cico 13323

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 ax-cnex 11163 ax-resscn 11164 ax-pre-lttri 11181 ax-pre-lttrn 11182

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-po 5588 df-so 5589 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-ov 7409 df-oprab 7410 df-mpo 7411 df-1st 7972 df-2nd 7973 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-ico 13327

This theorem is referenced by: fge0npnf 45070