icoub - Mathbox for Glauco Siliprandi

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Theorem icoub 39752

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 473	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ (𝐴[,)𝐵)) → 𝐴 ∈ ℝ^*)
2		icossxr 12258	. . . . 5 ⊢ (𝐴[,)𝐵) ⊆ ℝ^*
3		id 22	. . . . 5 ⊢ (𝐵 ∈ (𝐴[,)𝐵) → 𝐵 ∈ (𝐴[,)𝐵))
4	2, 3	sseldi 3601	. . . 4 ⊢ (𝐵 ∈ (𝐴[,)𝐵) → 𝐵 ∈ ℝ^*)
5	4	adantl 482	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ (𝐴[,)𝐵)) → 𝐵 ∈ ℝ^*)
6		simpr 477	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ (𝐴[,)𝐵)) → 𝐵 ∈ (𝐴[,)𝐵))
7		icoltub 39732	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐵 ∈ (𝐴[,)𝐵)) → 𝐵 < 𝐵)
8	1, 5, 6, 7	syl3anc 1326	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ (𝐴[,)𝐵)) → 𝐵 < 𝐵)
9		xrltnr 11953	. . . 4 ⊢ (𝐵 ∈ ℝ^* → ¬ 𝐵 < 𝐵)
10	4, 9	syl 17	. . 3 ⊢ (𝐵 ∈ (𝐴[,)𝐵) → ¬ 𝐵 < 𝐵)
11	10	adantl 482	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ (𝐴[,)𝐵)) → ¬ 𝐵 < 𝐵)
12	8, 11	pm2.65da 600	1 ⊢ (𝐴 ∈ ℝ^* → ¬ 𝐵 ∈ (𝐴[,)𝐵))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 384 ∈ wcel 1990 class class class wbr 4653 (class class class)co 6650 ℝ^*cxr 10073 < clt 10074 [,)cico 12177

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-pre-lttri 10010 ax-pre-lttrn 10011

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-po 5035 df-so 5036 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-ico 12181

This theorem is referenced by: fge0npnf 40584