icossico2d - Metamath Proof Explorer

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Theorem icossico2d 13469

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 13198	. 2 ⊢ (𝜑 → 𝐶 ≤ 𝐶)
5		icossico 13464	. 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 2146 ⊆ wss 3906 class class class wbr 5111 (class class class)co 7420 ℝ^*cxr 11262 ≤ cle 11264 [,)cico 13395

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 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7743 ax-cnex 11176 ax-resscn 11177 ax-pre-lttri 11194 ax-pre-lttrn 11195

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 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4875 df-iun 4960 df-br 5112 df-opab 5176 df-mpt 5195 df-id 5558 df-po 5571 df-so 5572 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-iota 6497 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-ov 7423 df-oprab 7424 df-mpo 7425 df-1st 7993 df-2nd 7994 df-er 8701 df-en 8951 df-dom 8952 df-sdom 8953 df-pnf 11265 df-mnf 11266 df-xr 11267 df-ltxr 11268 df-le 11269 df-ico 13399

This theorem is used by: liminflelimsuplem 46567 rehalfge1 48154

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