iocleub - Mathbox for Glauco Siliprandi

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Theorem iocleub 45121

Description: An element of a left-open right-closed interval is smaller than or equal to its upper bound. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

**Assertion**
Ref	Expression
iocleub	⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ (𝐴(,]𝐵)) → 𝐶 ≤ 𝐵)

**Proof of Theorem iocleub**
Step	Hyp	Ref	Expression
1		elioc1 13420	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐶 ∈ (𝐴(,]𝐵) ↔ (𝐶 ∈ ℝ^ ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵)))
2		simp3 1135	. . 3 ⊢ ((𝐶 ∈ ℝ^* ∧ 𝐴 < 𝐶 ∧ 𝐶 ≤ 𝐵) → 𝐶 ≤ 𝐵)
3	1, 2	biimtrdi 252	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐶 ∈ (𝐴(,]𝐵) → 𝐶 ≤ 𝐵))
4	3	3impia 1114	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ (𝐴(,]𝐵)) → 𝐶 ≤ 𝐵)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 ∧ w3a 1084 ∈ wcel 2099 class class class wbr 5153 (class class class)co 7424 ℝ^*cxr 11297 < clt 11298 ≤ cle 11299 (,]cioc 13379

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-resscn 11215

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-sbc 3777 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-br 5154 df-opab 5216 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-iota 6506 df-fun 6556 df-fv 6562 df-ov 7427 df-oprab 7428 df-mpo 7429 df-xr 11302 df-ioc 13383

This theorem is referenced by: iocopn 45138 iccdificc 45157 iocleubd 45177 limcresiooub 45263 fourierdlem19 45747 fourierdlem35 45763 fourierdlem41 45769 fourierdlem46 45773 fourierdlem48 45775 fourierdlem49 45776 fourierswlem 45851 fouriersw 45852 pimdecfgtioc 46336 pimincfltioc 46337

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