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Mathbox for Thierry Arnoux |
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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 versus 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 12622 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) → ((𝐴(,)𝐵) ∪ {𝐴, 𝐵}) = (𝐴[,]𝐵)) | |
2 | 1 | eleq2d 2845 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) → (𝐶 ∈ ((𝐴(,)𝐵) ∪ {𝐴, 𝐵}) ↔ 𝐶 ∈ (𝐴[,]𝐵))) |
3 | 2 | biimpar 471 | . . . 4 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) ∧ 𝐶 ∈ (𝐴[,]𝐵)) → 𝐶 ∈ ((𝐴(,)𝐵) ∪ {𝐴, 𝐵})) |
4 | elun 3976 | . . . . . 6 ⊢ (𝐶 ∈ ((𝐴(,)𝐵) ∪ {𝐴, 𝐵}) ↔ (𝐶 ∈ (𝐴(,)𝐵) ∨ 𝐶 ∈ {𝐴, 𝐵})) | |
5 | elprg 4419 | . . . . . . 7 ⊢ (𝐶 ∈ (𝐴[,]𝐵) → (𝐶 ∈ {𝐴, 𝐵} ↔ (𝐶 = 𝐴 ∨ 𝐶 = 𝐵))) | |
6 | 5 | orbi2d 902 | . . . . . 6 ⊢ (𝐶 ∈ (𝐴[,]𝐵) → ((𝐶 ∈ (𝐴(,)𝐵) ∨ 𝐶 ∈ {𝐴, 𝐵}) ↔ (𝐶 ∈ (𝐴(,)𝐵) ∨ (𝐶 = 𝐴 ∨ 𝐶 = 𝐵)))) |
7 | 4, 6 | syl5bb 275 | . . . . 5 ⊢ (𝐶 ∈ (𝐴[,]𝐵) → (𝐶 ∈ ((𝐴(,)𝐵) ∪ {𝐴, 𝐵}) ↔ (𝐶 ∈ (𝐴(,)𝐵) ∨ (𝐶 = 𝐴 ∨ 𝐶 = 𝐵)))) |
8 | 7 | adantl 475 | . . . 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 224 | . . 3 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) ∧ 𝐶 ∈ (𝐴[,]𝐵)) → (𝐶 ∈ (𝐴(,)𝐵) ∨ (𝐶 = 𝐴 ∨ 𝐶 = 𝐵))) |
10 | 3orass 1074 | . . . 4 ⊢ ((𝐶 ∈ (𝐴(,)𝐵) ∨ 𝐶 = 𝐴 ∨ 𝐶 = 𝐵) ↔ (𝐶 ∈ (𝐴(,)𝐵) ∨ (𝐶 = 𝐴 ∨ 𝐶 = 𝐵))) | |
11 | 3orcoma 1077 | . . . 4 ⊢ ((𝐶 ∈ (𝐴(,)𝐵) ∨ 𝐶 = 𝐴 ∨ 𝐶 = 𝐵) ↔ (𝐶 = 𝐴 ∨ 𝐶 ∈ (𝐴(,)𝐵) ∨ 𝐶 = 𝐵)) | |
12 | 10, 11 | bitr3i 269 | . . 3 ⊢ ((𝐶 ∈ (𝐴(,)𝐵) ∨ (𝐶 = 𝐴 ∨ 𝐶 = 𝐵)) ↔ (𝐶 = 𝐴 ∨ 𝐶 ∈ (𝐴(,)𝐵) ∨ 𝐶 = 𝐵)) |
13 | 9, 12 | sylib 210 | . 2 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) ∧ 𝐶 ∈ (𝐴[,]𝐵)) → (𝐶 = 𝐴 ∨ 𝐶 ∈ (𝐴(,)𝐵) ∨ 𝐶 = 𝐵)) |
14 | lbicc2 12606 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) → 𝐴 ∈ (𝐴[,]𝐵)) | |
15 | 14 | adantr 474 | . . . 4 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) ∧ 𝐶 = 𝐴) → 𝐴 ∈ (𝐴[,]𝐵)) |
16 | eleq1 2847 | . . . . 5 ⊢ (𝐶 = 𝐴 → (𝐶 ∈ (𝐴[,]𝐵) ↔ 𝐴 ∈ (𝐴[,]𝐵))) | |
17 | 16 | adantl 475 | . . . 4 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) ∧ 𝐶 = 𝐴) → (𝐶 ∈ (𝐴[,]𝐵) ↔ 𝐴 ∈ (𝐴[,]𝐵))) |
18 | 15, 17 | mpbird 249 | . . 3 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) ∧ 𝐶 = 𝐴) → 𝐶 ∈ (𝐴[,]𝐵)) |
19 | ioossicc 12575 | . . . . 5 ⊢ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵) | |
20 | 19 | sseli 3817 | . . . 4 ⊢ (𝐶 ∈ (𝐴(,)𝐵) → 𝐶 ∈ (𝐴[,]𝐵)) |
21 | 20 | adantl 475 | . . 3 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) ∧ 𝐶 ∈ (𝐴(,)𝐵)) → 𝐶 ∈ (𝐴[,]𝐵)) |
22 | ubicc2 12607 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) → 𝐵 ∈ (𝐴[,]𝐵)) | |
23 | 22 | adantr 474 | . . . 4 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) ∧ 𝐶 = 𝐵) → 𝐵 ∈ (𝐴[,]𝐵)) |
24 | eleq1 2847 | . . . . 5 ⊢ (𝐶 = 𝐵 → (𝐶 ∈ (𝐴[,]𝐵) ↔ 𝐵 ∈ (𝐴[,]𝐵))) | |
25 | 24 | adantl 475 | . . . 4 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) ∧ 𝐶 = 𝐵) → (𝐶 ∈ (𝐴[,]𝐵) ↔ 𝐵 ∈ (𝐴[,]𝐵))) |
26 | 23, 25 | mpbird 249 | . . 3 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) ∧ 𝐶 = 𝐵) → 𝐶 ∈ (𝐴[,]𝐵)) |
27 | 18, 21, 26 | 3jaodan 1504 | . 2 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) ∧ (𝐶 = 𝐴 ∨ 𝐶 ∈ (𝐴(,)𝐵) ∨ 𝐶 = 𝐵)) → 𝐶 ∈ (𝐴[,]𝐵)) |
28 | 13, 27 | impbida 791 | 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 198 ∧ wa 386 ∨ wo 836 ∨ w3o 1070 ∧ w3a 1071 = wceq 1601 ∈ wcel 2107 ∪ cun 3790 {cpr 4400 class class class wbr 4888 (class class class)co 6924 ℝ*cxr 10412 ≤ cle 10414 (,)cioo 12491 [,]cicc 12494 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-cnex 10330 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351 ax-pre-sup 10352 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-om 7346 df-1st 7447 df-2nd 7448 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-er 8028 df-en 8244 df-dom 8245 df-sdom 8246 df-sup 8638 df-inf 8639 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-div 11035 df-nn 11379 df-n0 11647 df-z 11733 df-uz 11997 df-q 12100 df-ioo 12495 df-ico 12497 df-icc 12498 |
This theorem is referenced by: elxrge02 30206 |
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