eliocd - Mathbox for Glauco Siliprandi

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Theorem eliocd 45062

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 13415	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐶 ∈ (𝐴(,]𝐵) ↔ (𝐶 ∈ ℝ^ ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵)))
7	4, 5, 6	syl2anc 582	. 2 ⊢ (𝜑 → (𝐶 ∈ (𝐴(,]𝐵) ↔ (𝐶 ∈ ℝ^* ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵)))
8	1, 2, 3, 7	mpbir3and 1339	1 ⊢ (𝜑 → 𝐶 ∈ (𝐴(,]𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ w3a 1084 ∈ wcel 2098 class class class wbr 5152 (class class class)co 7423 ℝ^*cxr 11293 < clt 11294 ≤ cle 11295 (,]cioc 13374

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5303 ax-nul 5310 ax-pr 5432 ax-un 7745 ax-cnex 11210 ax-resscn 11211

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-sbc 3776 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4325 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-id 5579 df-xp 5687 df-rel 5688 df-cnv 5689 df-co 5690 df-dm 5691 df-iota 6505 df-fun 6555 df-fv 6561 df-ov 7426 df-oprab 7427 df-mpo 7428 df-xr 11298 df-ioc 13378

This theorem is referenced by: iocopn 45075 eliccelioc 45076 iccdificc 45094 ressiocsup 45109 iooiinioc 45111 preimaiocmnf 45116 xlimpnfvlem2 45395 ioccncflimc 45443 fourierdlem41 45706 fourierdlem46 45710 fourierdlem48 45712 fourierdlem49 45713 fourierdlem51 45715 fourierswlem 45788 smfsuplem1 46369

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