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Mirrors > Home > MPE Home > Th. List > iccssioo2 | Structured version Visualization version GIF version |
Description: Condition for a closed interval to be a subset of an open interval. (Contributed by Mario Carneiro, 20-Feb-2015.) |
Ref | Expression |
---|---|
iccssioo2 | ⊢ ((𝐶 ∈ (𝐴(,)𝐵) ∧ 𝐷 ∈ (𝐴(,)𝐵)) → (𝐶[,]𝐷) ⊆ (𝐴(,)𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ne0i 3921 | . . . 4 ⊢ (𝐶 ∈ (𝐴(,)𝐵) → (𝐴(,)𝐵) ≠ ∅) | |
2 | 1 | adantr 481 | . . 3 ⊢ ((𝐶 ∈ (𝐴(,)𝐵) ∧ 𝐷 ∈ (𝐴(,)𝐵)) → (𝐴(,)𝐵) ≠ ∅) |
3 | ndmioo 12202 | . . . 4 ⊢ (¬ (𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴(,)𝐵) = ∅) | |
4 | 3 | necon1ai 2821 | . . 3 ⊢ ((𝐴(,)𝐵) ≠ ∅ → (𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*)) |
5 | 2, 4 | syl 17 | . 2 ⊢ ((𝐶 ∈ (𝐴(,)𝐵) ∧ 𝐷 ∈ (𝐴(,)𝐵)) → (𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*)) |
6 | eliooord 12233 | . . . 4 ⊢ (𝐶 ∈ (𝐴(,)𝐵) → (𝐴 < 𝐶 ∧ 𝐶 < 𝐵)) | |
7 | 6 | adantr 481 | . . 3 ⊢ ((𝐶 ∈ (𝐴(,)𝐵) ∧ 𝐷 ∈ (𝐴(,)𝐵)) → (𝐴 < 𝐶 ∧ 𝐶 < 𝐵)) |
8 | 7 | simpld 475 | . 2 ⊢ ((𝐶 ∈ (𝐴(,)𝐵) ∧ 𝐷 ∈ (𝐴(,)𝐵)) → 𝐴 < 𝐶) |
9 | eliooord 12233 | . . . 4 ⊢ (𝐷 ∈ (𝐴(,)𝐵) → (𝐴 < 𝐷 ∧ 𝐷 < 𝐵)) | |
10 | 9 | adantl 482 | . . 3 ⊢ ((𝐶 ∈ (𝐴(,)𝐵) ∧ 𝐷 ∈ (𝐴(,)𝐵)) → (𝐴 < 𝐷 ∧ 𝐷 < 𝐵)) |
11 | 10 | simprd 479 | . 2 ⊢ ((𝐶 ∈ (𝐴(,)𝐵) ∧ 𝐷 ∈ (𝐴(,)𝐵)) → 𝐷 < 𝐵) |
12 | iccssioo 12242 | . 2 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ (𝐴 < 𝐶 ∧ 𝐷 < 𝐵)) → (𝐶[,]𝐷) ⊆ (𝐴(,)𝐵)) | |
13 | 5, 8, 11, 12 | syl12anc 1324 | 1 ⊢ ((𝐶 ∈ (𝐴(,)𝐵) ∧ 𝐷 ∈ (𝐴(,)𝐵)) → (𝐶[,]𝐷) ⊆ (𝐴(,)𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 ∈ wcel 1990 ≠ wne 2794 ⊆ wss 3574 ∅c0 3915 class class class wbr 4653 (class class class)co 6650 ℝ*cxr 10073 < clt 10074 (,)cioo 12175 [,]cicc 12178 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-pre-lttri 10010 ax-pre-lttrn 10011 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-po 5035 df-so 5036 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-ioo 12179 df-icc 12182 |
This theorem is referenced by: dvivthlem1 23771 dvivthlem2 23772 amgmlem 24716 ioosconn 31229 amgmwlem 42548 |
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