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| Mirrors > Home > MPE Home > Th. List > iccssico2 | Structured version Visualization version GIF version | ||
| Description: Condition for a closed interval to be a subset of a closed-below, open-above interval. (Contributed by Mario Carneiro, 20-Feb-2015.) | 
| Ref | Expression | 
|---|---|
| iccssico2 | ⊢ ((𝐶 ∈ (𝐴[,)𝐵) ∧ 𝐷 ∈ (𝐴[,)𝐵)) → (𝐶[,]𝐷) ⊆ (𝐴[,)𝐵)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | df-ico 13393 | . . . 4 ⊢ [,) = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 < 𝑦)}) | |
| 2 | 1 | elmpocl1 7675 | . . 3 ⊢ (𝐶 ∈ (𝐴[,)𝐵) → 𝐴 ∈ ℝ*) | 
| 3 | 2 | adantr 480 | . 2 ⊢ ((𝐶 ∈ (𝐴[,)𝐵) ∧ 𝐷 ∈ (𝐴[,)𝐵)) → 𝐴 ∈ ℝ*) | 
| 4 | 1 | elmpocl2 7676 | . . 3 ⊢ (𝐶 ∈ (𝐴[,)𝐵) → 𝐵 ∈ ℝ*) | 
| 5 | 4 | adantr 480 | . 2 ⊢ ((𝐶 ∈ (𝐴[,)𝐵) ∧ 𝐷 ∈ (𝐴[,)𝐵)) → 𝐵 ∈ ℝ*) | 
| 6 | 1 | elixx3g 13400 | . . . . 5 ⊢ (𝐶 ∈ (𝐴[,)𝐵) ↔ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) ∧ (𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵))) | 
| 7 | 6 | simprbi 496 | . . . 4 ⊢ (𝐶 ∈ (𝐴[,)𝐵) → (𝐴 ≤ 𝐶 ∧ 𝐶 < 𝐵)) | 
| 8 | 7 | simpld 494 | . . 3 ⊢ (𝐶 ∈ (𝐴[,)𝐵) → 𝐴 ≤ 𝐶) | 
| 9 | 8 | adantr 480 | . 2 ⊢ ((𝐶 ∈ (𝐴[,)𝐵) ∧ 𝐷 ∈ (𝐴[,)𝐵)) → 𝐴 ≤ 𝐶) | 
| 10 | 1 | elixx3g 13400 | . . . . 5 ⊢ (𝐷 ∈ (𝐴[,)𝐵) ↔ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐷 ∈ ℝ*) ∧ (𝐴 ≤ 𝐷 ∧ 𝐷 < 𝐵))) | 
| 11 | 10 | simprbi 496 | . . . 4 ⊢ (𝐷 ∈ (𝐴[,)𝐵) → (𝐴 ≤ 𝐷 ∧ 𝐷 < 𝐵)) | 
| 12 | 11 | simprd 495 | . . 3 ⊢ (𝐷 ∈ (𝐴[,)𝐵) → 𝐷 < 𝐵) | 
| 13 | 12 | adantl 481 | . 2 ⊢ ((𝐶 ∈ (𝐴[,)𝐵) ∧ 𝐷 ∈ (𝐴[,)𝐵)) → 𝐷 < 𝐵) | 
| 14 | iccssico 13459 | . 2 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ (𝐴 ≤ 𝐶 ∧ 𝐷 < 𝐵)) → (𝐶[,]𝐷) ⊆ (𝐴[,)𝐵)) | |
| 15 | 3, 5, 9, 13, 14 | syl22anc 839 | 1 ⊢ ((𝐶 ∈ (𝐴[,)𝐵) ∧ 𝐷 ∈ (𝐴[,)𝐵)) → (𝐶[,]𝐷) ⊆ (𝐴[,)𝐵)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 ∈ wcel 2108 {crab 3436 ⊆ wss 3951 class class class wbr 5143 (class class class)co 7431 ℝ*cxr 11294 < clt 11295 ≤ cle 11296 [,)cico 13389 [,]cicc 13390 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-pre-lttri 11229 ax-pre-lttrn 11230 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-ico 13393 df-icc 13394 | 
| This theorem is referenced by: icopnfhmeo 24974 | 
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