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Mirrors > Home > MPE Home > Th. List > Mathboxes > eliccioo | Structured version Visualization version GIF version |
Description: Membership in a closed interval of extended reals vs. the same open interval. (Contributed by Thierry Arnoux, 18-Dec-2016.) |
Ref | Expression |
---|---|
eliccioo | ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐶 = 𝐴 ∨ 𝐶 ∈ (𝐴(,)𝐵) ∨ 𝐶 = 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prunioo 12301 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) → ((𝐴(,)𝐵) ∪ {𝐴, 𝐵}) = (𝐴[,]𝐵)) | |
2 | 1 | eleq2d 2687 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) → (𝐶 ∈ ((𝐴(,)𝐵) ∪ {𝐴, 𝐵}) ↔ 𝐶 ∈ (𝐴[,]𝐵))) |
3 | 2 | biimpar 502 | . . . 4 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) ∧ 𝐶 ∈ (𝐴[,]𝐵)) → 𝐶 ∈ ((𝐴(,)𝐵) ∪ {𝐴, 𝐵})) |
4 | elun 3753 | . . . . . 6 ⊢ (𝐶 ∈ ((𝐴(,)𝐵) ∪ {𝐴, 𝐵}) ↔ (𝐶 ∈ (𝐴(,)𝐵) ∨ 𝐶 ∈ {𝐴, 𝐵})) | |
5 | elprg 4196 | . . . . . . 7 ⊢ (𝐶 ∈ (𝐴[,]𝐵) → (𝐶 ∈ {𝐴, 𝐵} ↔ (𝐶 = 𝐴 ∨ 𝐶 = 𝐵))) | |
6 | 5 | orbi2d 738 | . . . . . 6 ⊢ (𝐶 ∈ (𝐴[,]𝐵) → ((𝐶 ∈ (𝐴(,)𝐵) ∨ 𝐶 ∈ {𝐴, 𝐵}) ↔ (𝐶 ∈ (𝐴(,)𝐵) ∨ (𝐶 = 𝐴 ∨ 𝐶 = 𝐵)))) |
7 | 4, 6 | syl5bb 272 | . . . . 5 ⊢ (𝐶 ∈ (𝐴[,]𝐵) → (𝐶 ∈ ((𝐴(,)𝐵) ∪ {𝐴, 𝐵}) ↔ (𝐶 ∈ (𝐴(,)𝐵) ∨ (𝐶 = 𝐴 ∨ 𝐶 = 𝐵)))) |
8 | 7 | adantl 482 | . . . 4 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) ∧ 𝐶 ∈ (𝐴[,]𝐵)) → (𝐶 ∈ ((𝐴(,)𝐵) ∪ {𝐴, 𝐵}) ↔ (𝐶 ∈ (𝐴(,)𝐵) ∨ (𝐶 = 𝐴 ∨ 𝐶 = 𝐵)))) |
9 | 3, 8 | mpbid 222 | . . 3 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) ∧ 𝐶 ∈ (𝐴[,]𝐵)) → (𝐶 ∈ (𝐴(,)𝐵) ∨ (𝐶 = 𝐴 ∨ 𝐶 = 𝐵))) |
10 | 3orass 1040 | . . . 4 ⊢ ((𝐶 ∈ (𝐴(,)𝐵) ∨ 𝐶 = 𝐴 ∨ 𝐶 = 𝐵) ↔ (𝐶 ∈ (𝐴(,)𝐵) ∨ (𝐶 = 𝐴 ∨ 𝐶 = 𝐵))) | |
11 | 3orcoma 1046 | . . . 4 ⊢ ((𝐶 ∈ (𝐴(,)𝐵) ∨ 𝐶 = 𝐴 ∨ 𝐶 = 𝐵) ↔ (𝐶 = 𝐴 ∨ 𝐶 ∈ (𝐴(,)𝐵) ∨ 𝐶 = 𝐵)) | |
12 | 10, 11 | bitr3i 266 | . . 3 ⊢ ((𝐶 ∈ (𝐴(,)𝐵) ∨ (𝐶 = 𝐴 ∨ 𝐶 = 𝐵)) ↔ (𝐶 = 𝐴 ∨ 𝐶 ∈ (𝐴(,)𝐵) ∨ 𝐶 = 𝐵)) |
13 | 9, 12 | sylib 208 | . 2 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) ∧ 𝐶 ∈ (𝐴[,]𝐵)) → (𝐶 = 𝐴 ∨ 𝐶 ∈ (𝐴(,)𝐵) ∨ 𝐶 = 𝐵)) |
14 | lbicc2 12288 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) → 𝐴 ∈ (𝐴[,]𝐵)) | |
15 | 14 | adantr 481 | . . . 4 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) ∧ 𝐶 = 𝐴) → 𝐴 ∈ (𝐴[,]𝐵)) |
16 | eleq1 2689 | . . . . 5 ⊢ (𝐶 = 𝐴 → (𝐶 ∈ (𝐴[,]𝐵) ↔ 𝐴 ∈ (𝐴[,]𝐵))) | |
17 | 16 | adantl 482 | . . . 4 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) ∧ 𝐶 = 𝐴) → (𝐶 ∈ (𝐴[,]𝐵) ↔ 𝐴 ∈ (𝐴[,]𝐵))) |
18 | 15, 17 | mpbird 247 | . . 3 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) ∧ 𝐶 = 𝐴) → 𝐶 ∈ (𝐴[,]𝐵)) |
19 | ioossicc 12259 | . . . . 5 ⊢ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵) | |
20 | 19 | sseli 3599 | . . . 4 ⊢ (𝐶 ∈ (𝐴(,)𝐵) → 𝐶 ∈ (𝐴[,]𝐵)) |
21 | 20 | adantl 482 | . . 3 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) ∧ 𝐶 ∈ (𝐴(,)𝐵)) → 𝐶 ∈ (𝐴[,]𝐵)) |
22 | ubicc2 12289 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) → 𝐵 ∈ (𝐴[,]𝐵)) | |
23 | 22 | adantr 481 | . . . 4 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) ∧ 𝐶 = 𝐵) → 𝐵 ∈ (𝐴[,]𝐵)) |
24 | eleq1 2689 | . . . . 5 ⊢ (𝐶 = 𝐵 → (𝐶 ∈ (𝐴[,]𝐵) ↔ 𝐵 ∈ (𝐴[,]𝐵))) | |
25 | 24 | adantl 482 | . . . 4 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) ∧ 𝐶 = 𝐵) → (𝐶 ∈ (𝐴[,]𝐵) ↔ 𝐵 ∈ (𝐴[,]𝐵))) |
26 | 23, 25 | mpbird 247 | . . 3 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) ∧ 𝐶 = 𝐵) → 𝐶 ∈ (𝐴[,]𝐵)) |
27 | 18, 21, 26 | 3jaodan 1394 | . 2 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) ∧ (𝐶 = 𝐴 ∨ 𝐶 ∈ (𝐴(,)𝐵) ∨ 𝐶 = 𝐵)) → 𝐶 ∈ (𝐴[,]𝐵)) |
28 | 13, 27 | impbida 877 | 1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐶 = 𝐴 ∨ 𝐶 ∈ (𝐴(,)𝐵) ∨ 𝐶 = 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∨ wo 383 ∧ wa 384 ∨ w3o 1036 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ∪ cun 3572 {cpr 4179 class class class wbr 4653 (class class class)co 6650 ℝ*cxr 10073 ≤ cle 10075 (,)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-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 |
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-reu 2919 df-rmo 2920 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-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 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-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 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-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-sup 8348 df-inf 8349 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-n0 11293 df-z 11378 df-uz 11688 df-q 11789 df-ioo 12179 df-ico 12181 df-icc 12182 |
This theorem is referenced by: elxrge02 29640 |
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