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Theorem iccssico2 13438

Description: Condition for a closed interval to be a subset of a closed-below, open-above interval. (Contributed by Mario Carneiro, 20-Feb-2015.)

**Assertion**
Ref	Expression
iccssico2	⊢ ((𝐶 ∈ (𝐴[,)𝐵) ∧ 𝐷 ∈ (𝐴[,)𝐵)) → (𝐶[,]𝐷) ⊆ (𝐴[,)𝐵))

Proof of Theorem iccssico2 Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ico 13369	. . . 4 ⊢ [,) = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)})
2	1	elmpocl1 7642	. . 3 ⊢ (𝐶 ∈ (𝐴[,)𝐵) → 𝐴 ∈ ℝ^*)
3	2	adantr 485	. 2 ⊢ ((𝐶 ∈ (𝐴[,)𝐵) ∧ 𝐷 ∈ (𝐴[,)𝐵)) → 𝐴 ∈ ℝ^*)
4	1	elmpocl2 7643	. . 3 ⊢ (𝐶 ∈ (𝐴[,)𝐵) → 𝐵 ∈ ℝ^*)
5	4	adantr 485	. 2 ⊢ ((𝐶 ∈ (𝐴[,)𝐵) ∧ 𝐷 ∈ (𝐴[,)𝐵)) → 𝐵 ∈ ℝ^*)
6	1	elixx3g 13376	. . . . 5 ⊢ (𝐶 ∈ (𝐴[,)𝐵) ↔ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵)))
7	6	simprbi 502	. . . 4 ⊢ (𝐶 ∈ (𝐴[,)𝐵) → (𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵))
8	7	simpld 499	. . 3 ⊢ (𝐶 ∈ (𝐴[,)𝐵) → 𝐴 ≤ 𝐶)
9	8	adantr 485	. 2 ⊢ ((𝐶 ∈ (𝐴[,)𝐵) ∧ 𝐷 ∈ (𝐴[,)𝐵)) → 𝐴 ≤ 𝐶)
10	1	elixx3g 13376	. . . . 5 ⊢ (𝐷 ∈ (𝐴[,)𝐵) ↔ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐷 ∈ ℝ^*) ∧ (𝐴 ≤ 𝐷 ∧ 𝐷 < 𝐵)))
11	10	simprbi 502	. . . 4 ⊢ (𝐷 ∈ (𝐴[,)𝐵) → (𝐴 ≤ 𝐷 ∧ 𝐷 < 𝐵))
12	11	simprd 500	. . 3 ⊢ (𝐷 ∈ (𝐴[,)𝐵) → 𝐷 < 𝐵)
13	12	adantl 486	. 2 ⊢ ((𝐶 ∈ (𝐴[,)𝐵) ∧ 𝐷 ∈ (𝐴[,)𝐵)) → 𝐷 < 𝐵)
14		iccssico 13436	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ (𝐴 ≤ 𝐶 ∧ 𝐷 < 𝐵)) → (𝐶[,]𝐷) ⊆ (𝐴[,)𝐵))
15	3, 5, 9, 13, 14	syl22anc 851	1 ⊢ ((𝐶 ∈ (𝐴[,)𝐵) ∧ 𝐷 ∈ (𝐴[,)𝐵)) → (𝐶[,]𝐷) ⊆ (𝐴[,)𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 400 ∧ w3a 1101 ∈ wcel 2145 {crab 3417 ⊆ wss 3907 class class class wbr 5105 (class class class)co 7400 ℝ^*cxr 11230 < clt 11231 ≤ cle 11232 [,)cico 13365 [,]cicc 13366

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1818 ax-4 1832 ax-5 1933 ax-6 1990 ax-7 2031 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2737 ax-sep 5251 ax-nul 5261 ax-pow 5327 ax-pr 5395 ax-un 7722 ax-cnex 11144 ax-resscn 11145 ax-pre-lttri 11162 ax-pre-lttrn 11163

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1566 df-fal 1576 df-ex 1803 df-nf 1807 df-sb 2094 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3065 df-ral 3080 df-rex 3090 df-rab 3418 df-v 3459 df-sbc 3748 df-csb 3856 df-dif 3910 df-un 3912 df-in 3914 df-ss 3924 df-nul 4289 df-if 4484 df-pw 4560 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4869 df-iun 4954 df-br 5106 df-opab 5168 df-mpt 5187 df-id 5547 df-po 5560 df-so 5561 df-xp 5658 df-rel 5659 df-cnv 5660 df-co 5661 df-dm 5662 df-rn 5663 df-res 5664 df-ima 5665 df-iota 6481 df-fun 6527 df-fn 6528 df-f 6529 df-f1 6530 df-fo 6531 df-f1o 6532 df-fv 6533 df-ov 7403 df-oprab 7404 df-mpo 7405 df-1st 7974 df-2nd 7975 df-er 8682 df-en 8932 df-dom 8933 df-sdom 8934 df-pnf 11233 df-mnf 11234 df-xr 11235 df-ltxr 11236 df-le 11237 df-ico 13369 df-icc 13370

This theorem is referenced by: icopnfhmeo 25063

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