eliccelicod - Mathbox for Glauco Siliprandi

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Theorem eliccelicod 44828

Description: A member of a closed interval that is not the upper bound, is a member of the left-closed, right-open interval. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

**Hypotheses**
Ref	Expression
eliccelicod.a	⊢ (𝜑 → 𝐴 ∈ ℝ^*)
eliccelicod.b	⊢ (𝜑 → 𝐵 ∈ ℝ^*)
eliccelicod.c	⊢ (𝜑 → 𝐶 ∈ (𝐴[,]𝐵))
eliccelicod.d	⊢ (𝜑 → 𝐶 ≠ 𝐵)

**Assertion**
Ref	Expression
eliccelicod	⊢ (𝜑 → 𝐶 ∈ (𝐴[,)𝐵))

**Proof of Theorem eliccelicod**
Step	Hyp	Ref	Expression
1		eliccelicod.a	. 2 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
2		eliccelicod.b	. 2 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
3		eliccelicod.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ (𝐴[,]𝐵))
4		eliccxr 13430	. . 3 ⊢ (𝐶 ∈ (𝐴[,]𝐵) → 𝐶 ∈ ℝ^*)
5	3, 4	syl 17	. 2 ⊢ (𝜑 → 𝐶 ∈ ℝ^*)
6		iccgelb 13398	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ (𝐴[,]𝐵)) → 𝐴 ≤ 𝐶)
7	1, 2, 3, 6	syl3anc 1369	. 2 ⊢ (𝜑 → 𝐴 ≤ 𝐶)
8		iccleub 13397	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ (𝐴[,]𝐵)) → 𝐶 ≤ 𝐵)
9	1, 2, 3, 8	syl3anc 1369	. . 3 ⊢ (𝜑 → 𝐶 ≤ 𝐵)
10		eliccelicod.d	. . 3 ⊢ (𝜑 → 𝐶 ≠ 𝐵)
11	5, 2, 9, 10	xrleneltd 44618	. 2 ⊢ (𝜑 → 𝐶 < 𝐵)
12	1, 2, 5, 7, 11	elicod 13392	1 ⊢ (𝜑 → 𝐶 ∈ (𝐴[,)𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∈ wcel 2099 ≠ wne 2935 class class class wbr 5142 (class class class)co 7414 ℝ^*cxr 11263 ≤ cle 11265 [,)cico 13344 [,]cicc 13345

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 ax-cnex 11180 ax-resscn 11181 ax-pre-lttri 11198 ax-pre-lttrn 11199

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-po 5584 df-so 5585 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-ov 7417 df-oprab 7418 df-mpo 7419 df-1st 7985 df-2nd 7986 df-er 8716 df-en 8954 df-dom 8955 df-sdom 8956 df-pnf 11266 df-mnf 11267 df-xr 11268 df-ltxr 11269 df-le 11270 df-ico 13348 df-icc 13349

This theorem is referenced by: carageniuncl 45824

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