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Mirrors > Home > MPE Home > Th. List > iccss | Structured version Visualization version GIF version |
Description: Condition for a closed interval to be a subset of another closed interval. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 20-Feb-2015.) |
Ref | Expression |
---|---|
iccss | ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐴 ≤ 𝐶 ∧ 𝐷 ≤ 𝐵)) → (𝐶[,]𝐷) ⊆ (𝐴[,]𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rexr 10879 | . . 3 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ*) | |
2 | rexr 10879 | . . 3 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℝ*) | |
3 | 1, 2 | anim12i 616 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*)) |
4 | df-icc 12942 | . . 3 ⊢ [,] = (𝑥 ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (𝑥 ≤ 𝑧 ∧ 𝑧 ≤ 𝑦)}) | |
5 | xrletr 12748 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ∧ 𝑤 ∈ ℝ*) → ((𝐴 ≤ 𝐶 ∧ 𝐶 ≤ 𝑤) → 𝐴 ≤ 𝑤)) | |
6 | xrletr 12748 | . . 3 ⊢ ((𝑤 ∈ ℝ* ∧ 𝐷 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → ((𝑤 ≤ 𝐷 ∧ 𝐷 ≤ 𝐵) → 𝑤 ≤ 𝐵)) | |
7 | 4, 4, 5, 6 | ixxss12 12955 | . 2 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ (𝐴 ≤ 𝐶 ∧ 𝐷 ≤ 𝐵)) → (𝐶[,]𝐷) ⊆ (𝐴[,]𝐵)) |
8 | 3, 7 | sylan 583 | 1 ⊢ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ (𝐴 ≤ 𝐶 ∧ 𝐷 ≤ 𝐵)) → (𝐶[,]𝐷) ⊆ (𝐴[,]𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∈ wcel 2110 ⊆ wss 3866 class class class wbr 5053 (class class class)co 7213 ℝcr 10728 ℝ*cxr 10866 ≤ cle 10868 [,]cicc 12938 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-pre-lttri 10803 ax-pre-lttrn 10804 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-id 5455 df-po 5468 df-so 5469 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-ov 7216 df-oprab 7217 df-mpo 7218 df-1st 7761 df-2nd 7762 df-er 8391 df-en 8627 df-dom 8628 df-sdom 8629 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-icc 12942 |
This theorem is referenced by: xrhmeo 23843 lebnumii 23863 pcoval1 23910 pcoval2 23913 ivthicc 24355 dyaddisjlem 24492 volsup2 24502 volcn 24503 mbfi1fseqlem5 24617 dvcvx 24917 dvfsumle 24918 dvfsumabs 24920 harmonicbnd3 25890 ppisval 25986 chtwordi 26038 ppiwordi 26044 chpub 26101 cvmliftlem2 32961 fourierdlem76 43398 fourierdlem103 43425 fourierdlem104 43426 fourierdlem107 43429 fourierdlem112 43434 salexct3 43556 salgensscntex 43558 icccldii 45885 sepfsepc 45894 |
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