icossico2 - Mathbox for Glauco Siliprandi

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Theorem icossico2 41382

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
icossico2.1	⊢ (𝜑 → 𝐵 ∈ ℝ^*)
icossico2.2	⊢ (𝜑 → 𝐶 ∈ ℝ^*)
icossico2.3	⊢ (𝜑 → 𝐵 ≤ 𝐴)

**Assertion**
Ref	Expression
icossico2	⊢ (𝜑 → (𝐴[,)𝐶) ⊆ (𝐵[,)𝐶))

**Proof of Theorem icossico2**
Step	Hyp	Ref	Expression
1		icossico2.1	. 2 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
2		icossico2.2	. 2 ⊢ (𝜑 → 𝐶 ∈ ℝ^*)
3		icossico2.3	. 2 ⊢ (𝜑 → 𝐵 ≤ 𝐴)
4	2	xrleidd 12395	. 2 ⊢ (𝜑 → 𝐶 ≤ 𝐶)
5		icossico 12656	. 2 ⊢ (((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝐵 ≤ 𝐴 ∧ 𝐶 ≤ 𝐶)) → (𝐴[,)𝐶) ⊆ (𝐵[,)𝐶))
6	1, 2, 3, 4, 5	syl22anc 835	1 ⊢ (𝜑 → (𝐴[,)𝐶) ⊆ (𝐵[,)𝐶))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∈ wcel 2081 ⊆ wss 3859 class class class wbr 4962 (class class class)co 7016 ℝ^*cxr 10520 ≤ cle 10522 [,)cico 12590

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 ax-cnex 10439 ax-resscn 10440 ax-pre-lttri 10457 ax-pre-lttrn 10458

This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-op 4479 df-uni 4746 df-iun 4827 df-br 4963 df-opab 5025 df-mpt 5042 df-id 5348 df-po 5362 df-so 5363 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-fv 6233 df-ov 7019 df-oprab 7020 df-mpo 7021 df-1st 7545 df-2nd 7546 df-er 8139 df-en 8358 df-dom 8359 df-sdom 8360 df-pnf 10523 df-mnf 10524 df-xr 10525 df-ltxr 10526 df-le 10527 df-ico 12594

This theorem is referenced by: liminflelimsuplem 41598

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