eliccelicod - Mathbox for Glauco Siliprandi

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Theorem eliccelicod 45981

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 13382	. . 3 ⊢ (𝐶 ∈ (𝐴[,]𝐵) → 𝐶 ∈ ℝ^*)
5	3, 4	syl 17	. 2 ⊢ (𝜑 → 𝐶 ∈ ℝ^*)
6		iccgelb 13349	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ (𝐴[,]𝐵)) → 𝐴 ≤ 𝐶)
7	1, 2, 3, 6	syl3anc 1374	. 2 ⊢ (𝜑 → 𝐴 ≤ 𝐶)
8		iccleub 13348	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ (𝐴[,]𝐵)) → 𝐶 ≤ 𝐵)
9	1, 2, 3, 8	syl3anc 1374	. . 3 ⊢ (𝜑 → 𝐶 ≤ 𝐵)
10		eliccelicod.d	. . 3 ⊢ (𝜑 → 𝐶 ≠ 𝐵)
11	5, 2, 9, 10	xrleneltd 45774	. 2 ⊢ (𝜑 → 𝐶 < 𝐵)
12	1, 2, 5, 7, 11	elicod 13342	1 ⊢ (𝜑 → 𝐶 ∈ (𝐴[,)𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∈ wcel 2114 ≠ wne 2933 class class class wbr 5086 (class class class)co 7361 ℝ^*cxr 11172 ≤ cle 11174 [,)cico 13294 [,]cicc 13295

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5232 ax-nul 5242 ax-pow 5303 ax-pr 5371 ax-un 7683 ax-cnex 11088 ax-resscn 11089 ax-pre-lttri 11106 ax-pre-lttrn 11107

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-id 5520 df-po 5533 df-so 5534 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-ov 7364 df-oprab 7365 df-mpo 7366 df-1st 7936 df-2nd 7937 df-er 8637 df-en 8888 df-dom 8889 df-sdom 8890 df-pnf 11175 df-mnf 11176 df-xr 11177 df-ltxr 11178 df-le 11179 df-ico 13298 df-icc 13299

This theorem is referenced by: carageniuncl 46972

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