icossico2d - Metamath Proof Explorer

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Theorem icossico2d 13499

Description: Condition for a closed-below, open-above interval to be a subset of a closed-below, open-above interval. (Contributed by Glauco Siliprandi, 2-Jan-2022.)

**Hypotheses**
Ref	Expression
icossico2d.1	⊢ (𝜑 → 𝐵 ∈ ℝ^*)
icossico2d.2	⊢ (𝜑 → 𝐶 ∈ ℝ^*)
icossico2d.3	⊢ (𝜑 → 𝐵 ≤ 𝐴)

**Assertion**
Ref	Expression
icossico2d	⊢ (𝜑 → (𝐴[,)𝐶) ⊆ (𝐵[,)𝐶))

**Proof of Theorem icossico2d**
Step	Hyp	Ref	Expression
1		icossico2d.1	. 2 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
2		icossico2d.2	. 2 ⊢ (𝜑 → 𝐶 ∈ ℝ^*)
3		icossico2d.3	. 2 ⊢ (𝜑 → 𝐵 ≤ 𝐴)
4	2	xrleidd 13228	. 2 ⊢ (𝜑 → 𝐶 ≤ 𝐶)
5		icossico 13494	. 2 ⊢ (((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝐵 ≤ 𝐴 ∧ 𝐶 ≤ 𝐶)) → (𝐴[,)𝐶) ⊆ (𝐵[,)𝐶))
6	1, 2, 3, 4, 5	syl22anc 852	1 ⊢ (𝜑 → (𝐴[,)𝐶) ⊆ (𝐵[,)𝐶))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∈ wcel 2145 ⊆ wss 3899 class class class wbr 5103 (class class class)co 7416 ℝ^*cxr 11291 ≤ cle 11293 [,)cico 13425

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7742 ax-cnex 11205 ax-resscn 11206 ax-pre-lttri 11223 ax-pre-lttrn 11224

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-id 5550 df-po 5563 df-so 5564 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-iota 6491 df-fun 6537 df-fn 6538 df-f 6539 df-f1 6540 df-fo 6541 df-f1o 6542 df-fv 6543 df-ov 7419 df-oprab 7420 df-mpo 7421 df-1st 7992 df-2nd 7993 df-er 8703 df-en 8960 df-dom 8961 df-sdom 8962 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-ico 13429

This theorem is used by: liminflelimsuplem 46668 rehalfge1 48292

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