eliocd - Mathbox for Glauco Siliprandi

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Theorem eliocd 45505

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 13348	. . 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 1086 ∈ wcel 2109 class class class wbr 5107 (class class class)co 7387 ℝ^*cxr 11207 < clt 11208 ≤ cle 11209 (,]cioc 13307

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5251 ax-nul 5261 ax-pr 5387 ax-un 7711 ax-cnex 11124 ax-resscn 11125

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ral 3045 df-rex 3054 df-rab 3406 df-v 3449 df-sbc 3754 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-br 5108 df-opab 5170 df-id 5533 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-iota 6464 df-fun 6513 df-fv 6519 df-ov 7390 df-oprab 7391 df-mpo 7392 df-xr 11212 df-ioc 13311

This theorem is referenced by: iocopn 45518 eliccelioc 45519 iccdificc 45537 ressiocsup 45552 iooiinioc 45554 preimaiocmnf 45558 xlimpnfvlem2 45835 ioccncflimc 45883 fourierdlem41 46146 fourierdlem46 46150 fourierdlem48 46152 fourierdlem49 46153 fourierdlem51 46155 fourierswlem 46228 smfsuplem1 46809

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