eliocd - Mathbox for Glauco Siliprandi

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Theorem eliocd 46612

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 13521	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐶 ∈ (𝐴(,]𝐵) ↔ (𝐶 ∈ ℝ^ ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵)))
7	4, 5, 6	syl2anc 596	. 2 ⊢ (𝜑 → (𝐶 ∈ (𝐴(,]𝐵) ↔ (𝐶 ∈ ℝ^* ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵)))
8	1, 2, 3, 7	mpbir3and 1361	1 ⊢ (𝜑 → 𝐶 ∈ (𝐴(,]𝐵))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ w3a 1103 ∈ wcel 2144 class class class wbr 5102 (class class class)co 7420 ℝ^*cxr 11345 < clt 11346 ≤ cle 11347 (,]cioc 13480

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-sep 5248 ax-pr 5390 ax-un 7751 ax-cnex 11259 ax-resscn 11260

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ral 3077 df-rex 3087 df-rab 3413 df-v 3452 df-sbc 3739 df-ss 3896 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-op 4835 df-uni 4867 df-br 5103 df-opab 5167 df-id 5545 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-iota 6494 df-fun 6540 df-fv 6546 df-ov 7423 df-oprab 7424 df-mpo 7425 df-xr 11350 df-ioc 13484

This theorem is used by: iocopn 46625 eliccelioc 46626 iccdificc 46644 ressiocsup 46659 iooiinioc 46661 preimaiocmnf 46665 xlimpnfvlem2 46940 ioccncflimc 46988 fourierdlem41 47251 fourierdlem46 47255 fourierdlem48 47257 fourierdlem49 47258 fourierdlem51 47260 fourierswlem 47333 smfsuplem1 47914

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