eliccelicod - Mathbox for Glauco Siliprandi

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Theorem eliccelicod 41826

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 12824	. . 3 ⊢ (𝐶 ∈ (𝐴[,]𝐵) → 𝐶 ∈ ℝ^*)
5	3, 4	syl 17	. 2 ⊢ (𝜑 → 𝐶 ∈ ℝ^*)
6		iccgelb 12794	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ (𝐴[,]𝐵)) → 𝐴 ≤ 𝐶)
7	1, 2, 3, 6	syl3anc 1367	. 2 ⊢ (𝜑 → 𝐴 ≤ 𝐶)
8		iccleub 12793	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ (𝐴[,]𝐵)) → 𝐶 ≤ 𝐵)
9	1, 2, 3, 8	syl3anc 1367	. . 3 ⊢ (𝜑 → 𝐶 ≤ 𝐵)
10		eliccelicod.d	. . 3 ⊢ (𝜑 → 𝐶 ≠ 𝐵)
11	5, 2, 9, 10	xrleneltd 41611	. 2 ⊢ (𝜑 → 𝐶 < 𝐵)
12	1, 2, 5, 7, 11	elicod 12788	1 ⊢ (𝜑 → 𝐶 ∈ (𝐴[,)𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∈ wcel 2114 ≠ wne 3016 class class class wbr 5066 (class class class)co 7156 ℝ^*cxr 10674 ≤ cle 10676 [,)cico 12741 [,]cicc 12742

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-pre-lttri 10611 ax-pre-lttrn 10612

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-po 5474 df-so 5475 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-1st 7689 df-2nd 7690 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-ico 12745 df-icc 12746

This theorem is referenced by: carageniuncl 42825

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