eliocd - Mathbox for Glauco Siliprandi

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Theorem eliocd 45520

Description: Membership in a left-open right-closed interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

**Hypotheses**
Ref	Expression
eliocd.a	⊢ (𝜑 → 𝐴 ∈ ℝ^*)
eliocd.b	⊢ (𝜑 → 𝐵 ∈ ℝ^*)
eliocd.c	⊢ (𝜑 → 𝐶 ∈ ℝ^*)
eliocd.altc	⊢ (𝜑 → 𝐴 < 𝐶)
eliocd.cleb	⊢ (𝜑 → 𝐶 ≤ 𝐵)

**Assertion**
Ref	Expression
eliocd	⊢ (𝜑 → 𝐶 ∈ (𝐴(,]𝐵))

**Proof of Theorem eliocd**
Step	Hyp	Ref	Expression
1		eliocd.c	. 2 ⊢ (𝜑 → 𝐶 ∈ ℝ^*)
2		eliocd.altc	. 2 ⊢ (𝜑 → 𝐴 < 𝐶)
3		eliocd.cleb	. 2 ⊢ (𝜑 → 𝐶 ≤ 𝐵)
4		eliocd.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
5		eliocd.b	. . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
6		elioc1 13429	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐶 ∈ (𝐴(,]𝐵) ↔ (𝐶 ∈ ℝ^ ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵)))
7	4, 5, 6	syl2anc 584	. 2 ⊢ (𝜑 → (𝐶 ∈ (𝐴(,]𝐵) ↔ (𝐶 ∈ ℝ^* ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵)))
8	1, 2, 3, 7	mpbir3and 1343	1 ⊢ (𝜑 → 𝐶 ∈ (𝐴(,]𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ w3a 1087 ∈ wcel 2108 class class class wbr 5143 (class class class)co 7431 ℝ^*cxr 11294 < clt 11295 ≤ cle 11296 (,]cioc 13388

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-xr 11299 df-ioc 13392

This theorem is referenced by: iocopn 45533 eliccelioc 45534 iccdificc 45552 ressiocsup 45567 iooiinioc 45569 preimaiocmnf 45574 xlimpnfvlem2 45852 ioccncflimc 45900 fourierdlem41 46163 fourierdlem46 46167 fourierdlem48 46169 fourierdlem49 46170 fourierdlem51 46172 fourierswlem 46245 smfsuplem1 46826

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