icoub - Mathbox for Glauco Siliprandi

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Theorem icoub 46645

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 488	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ (𝐴[,)𝐵)) → 𝐴 ∈ ℝ^*)
2		icossxr 13568	. . . . 5 ⊢ (𝐴[,)𝐵) ⊆ ℝ^*
3		id 23	. . . . 5 ⊢ (𝐵 ∈ (𝐴[,)𝐵) → 𝐵 ∈ (𝐴[,)𝐵))
4	2, 3	sselid 3907	. . . 4 ⊢ (𝐵 ∈ (𝐴[,)𝐵) → 𝐵 ∈ ℝ^*)
5	4	adantl 487	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ (𝐴[,)𝐵)) → 𝐵 ∈ ℝ^*)
6		simpr 490	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ (𝐴[,)𝐵)) → 𝐵 ∈ (𝐴[,)𝐵))
7		icoltub 46627	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐵 ∈ (𝐴[,)𝐵)) → 𝐵 < 𝐵)
8	1, 5, 6, 7	syl3anc 1398	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ (𝐴[,)𝐵)) → 𝐵 < 𝐵)
9		xrltnr 13253	. . . 4 ⊢ (𝐵 ∈ ℝ^* → ¬ 𝐵 < 𝐵)
10	4, 9	syl 18	. . 3 ⊢ (𝐵 ∈ (𝐴[,)𝐵) → ¬ 𝐵 < 𝐵)
11	10	adantl 487	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ (𝐴[,)𝐵)) → ¬ 𝐵 < 𝐵)
12	8, 11	pm2.65da 829	1 ⊢ (𝐴 ∈ ℝ^* → ¬ 𝐵 ∈ (𝐴[,)𝐵))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 401 ∈ wcel 2144 class class class wbr 5102 (class class class)co 7422 ℝ^*cxr 11347 < clt 11348 [,)cico 13483

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7753 ax-cnex 11261 ax-resscn 11262 ax-pre-lttri 11279 ax-pre-lttrn 11280

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3848 df-ss 3897 df-dif 4046 df-in 4090 df-un 4150 df-nul 4278 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-op 4835 df-uni 4867 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-id 5545 df-po 5558 df-so 5559 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-iota 6496 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-ov 7425 df-oprab 7426 df-mpo 7427 df-1st 8003 df-2nd 8004 df-er 8722 df-en 8979 df-dom 8980 df-sdom 8981 df-pnf 11350 df-mnf 11351 df-xr 11352 df-ltxr 11353 df-ico 13487

This theorem is used by: fge0npnf 47484

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