icossico2d - Metamath Proof Explorer

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Theorem icossico2d 13576

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 13305	. 2 ⊢ (𝜑 → 𝐶 ≤ 𝐶)
5		icossico 13571	. 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 2147 ⊆ wss 3900 class class class wbr 5106 (class class class)co 7440 ℝ^*cxr 11366 ≤ cle 11368 [,)cico 13502

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 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-sep 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395 ax-un 7772 ax-cnex 11280 ax-resscn 11281 ax-pre-lttri 11298 ax-pre-lttrn 11299

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 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4839 df-uni 4871 df-iun 4956 df-br 5107 df-opab 5171 df-mpt 5190 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 6509 df-fun 6557 df-fn 6558 df-f 6559 df-f1 6560 df-fo 6561 df-f1o 6562 df-fv 6563 df-ov 7443 df-oprab 7444 df-mpo 7445 df-1st 8022 df-2nd 8023 df-er 8742 df-en 8999 df-dom 9000 df-sdom 9001 df-pnf 11369 df-mnf 11370 df-xr 11371 df-ltxr 11372 df-le 11373 df-ico 13506

This theorem is used by: liminflelimsuplem 46988 rehalfge1 48612

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