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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 13524 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) → ((𝐴(,)𝐵) ∪ {𝐴, 𝐵}) = (𝐴[,]𝐵)) | |
| 2 | 1 | eleq2d 2851 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) → (𝐶 ∈ ((𝐴(,)𝐵) ∪ {𝐴, 𝐵}) ↔ 𝐶 ∈ (𝐴[,]𝐵))) |
| 3 | 2 | biimpar 483 | . . . 4 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) ∧ 𝐶 ∈ (𝐴[,]𝐵)) → 𝐶 ∈ ((𝐴(,)𝐵) ∪ {𝐴, 𝐵})) |
| 4 | elun 4107 | . . . . . 6 ⊢ (𝐶 ∈ ((𝐴(,)𝐵) ∪ {𝐴, 𝐵}) ↔ (𝐶 ∈ (𝐴(,)𝐵) ∨ 𝐶 ∈ {𝐴, 𝐵})) | |
| 5 | elprg 4614 | . . . . . . 7 ⊢ (𝐶 ∈ (𝐴[,]𝐵) → (𝐶 ∈ {𝐴, 𝐵} ↔ (𝐶 = 𝐴 ∨ 𝐶 = 𝐵))) | |
| 6 | 5 | orbi2d 929 | . . . . . 6 ⊢ (𝐶 ∈ (𝐴[,]𝐵) → ((𝐶 ∈ (𝐴(,)𝐵) ∨ 𝐶 ∈ {𝐴, 𝐵}) ↔ (𝐶 ∈ (𝐴(,)𝐵) ∨ (𝐶 = 𝐴 ∨ 𝐶 = 𝐵)))) |
| 7 | 4, 6 | bitrid 286 | . . . . 5 ⊢ (𝐶 ∈ (𝐴[,]𝐵) → (𝐶 ∈ ((𝐴(,)𝐵) ∪ {𝐴, 𝐵}) ↔ (𝐶 ∈ (𝐴(,)𝐵) ∨ (𝐶 = 𝐴 ∨ 𝐶 = 𝐵)))) |
| 8 | 7 | adantl 487 | . . . 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 235 | . . 3 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) ∧ 𝐶 ∈ (𝐴[,]𝐵)) → (𝐶 ∈ (𝐴(,)𝐵) ∨ (𝐶 = 𝐴 ∨ 𝐶 = 𝐵))) |
| 10 | 3orass 1106 | . . . 4 ⊢ ((𝐶 ∈ (𝐴(,)𝐵) ∨ 𝐶 = 𝐴 ∨ 𝐶 = 𝐵) ↔ (𝐶 ∈ (𝐴(,)𝐵) ∨ (𝐶 = 𝐴 ∨ 𝐶 = 𝐵))) | |
| 11 | 3orcoma 1109 | . . . 4 ⊢ ((𝐶 ∈ (𝐴(,)𝐵) ∨ 𝐶 = 𝐴 ∨ 𝐶 = 𝐵) ↔ (𝐶 = 𝐴 ∨ 𝐶 ∈ (𝐴(,)𝐵) ∨ 𝐶 = 𝐵)) | |
| 12 | 10, 11 | bitr3i 280 | . . 3 ⊢ ((𝐶 ∈ (𝐴(,)𝐵) ∨ (𝐶 = 𝐴 ∨ 𝐶 = 𝐵)) ↔ (𝐶 = 𝐴 ∨ 𝐶 ∈ (𝐴(,)𝐵) ∨ 𝐶 = 𝐵)) |
| 13 | 9, 12 | sylib 221 | . 2 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) ∧ 𝐶 ∈ (𝐴[,]𝐵)) → (𝐶 = 𝐴 ∨ 𝐶 ∈ (𝐴(,)𝐵) ∨ 𝐶 = 𝐵)) |
| 14 | lbicc2 13507 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) → 𝐴 ∈ (𝐴[,]𝐵)) | |
| 15 | 14 | adantr 486 | . . . 4 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) ∧ 𝐶 = 𝐴) → 𝐴 ∈ (𝐴[,]𝐵)) |
| 16 | eleq1 2853 | . . . . 5 ⊢ (𝐶 = 𝐴 → (𝐶 ∈ (𝐴[,]𝐵) ↔ 𝐴 ∈ (𝐴[,]𝐵))) | |
| 17 | 16 | adantl 487 | . . . 4 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) ∧ 𝐶 = 𝐴) → (𝐶 ∈ (𝐴[,]𝐵) ↔ 𝐴 ∈ (𝐴[,]𝐵))) |
| 18 | 15, 17 | mpbird 260 | . . 3 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) ∧ 𝐶 = 𝐴) → 𝐶 ∈ (𝐴[,]𝐵)) |
| 19 | ioossicc 13476 | . . . . 5 ⊢ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵) | |
| 20 | 19 | sseli 3934 | . . . 4 ⊢ (𝐶 ∈ (𝐴(,)𝐵) → 𝐶 ∈ (𝐴[,]𝐵)) |
| 21 | 20 | adantl 487 | . . 3 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) ∧ 𝐶 ∈ (𝐴(,)𝐵)) → 𝐶 ∈ (𝐴[,]𝐵)) |
| 22 | ubicc2 13508 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) → 𝐵 ∈ (𝐴[,]𝐵)) | |
| 23 | 22 | adantr 486 | . . . 4 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) ∧ 𝐶 = 𝐵) → 𝐵 ∈ (𝐴[,]𝐵)) |
| 24 | eleq1 2853 | . . . . 5 ⊢ (𝐶 = 𝐵 → (𝐶 ∈ (𝐴[,]𝐵) ↔ 𝐵 ∈ (𝐴[,]𝐵))) | |
| 25 | 24 | adantl 487 | . . . 4 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) ∧ 𝐶 = 𝐵) → (𝐶 ∈ (𝐴[,]𝐵) ↔ 𝐵 ∈ (𝐴[,]𝐵))) |
| 26 | 23, 25 | mpbird 260 | . . 3 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) ∧ 𝐶 = 𝐵) → 𝐶 ∈ (𝐴[,]𝐵)) |
| 27 | 18, 21, 26 | 3jaodan 1458 | . 2 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) ∧ (𝐶 = 𝐴 ∨ 𝐶 ∈ (𝐴(,)𝐵) ∨ 𝐶 = 𝐵)) → 𝐶 ∈ (𝐴[,]𝐵)) |
| 28 | 13, 27 | impbida 813 | 1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) → (𝐶 ∈ (𝐴[,]𝐵) ↔ (𝐶 = 𝐴 ∨ 𝐶 ∈ (𝐴(,)𝐵) ∨ 𝐶 = 𝐵))) |
| Colors of variables: wff setvar class |
| This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∨ wo 861 ∨ w3o 1102 ∧ w3a 1103 = wceq 1570 ∈ wcel 2146 ∪ cun 3904 {cpr 4593 class class class wbr 5111 (class class class)co 7419 ℝ*cxr 11257 ≤ cle 11259 (,)cioo 13388 [,]cicc 13391 |
| This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7742 ax-cnex 11171 ax-resscn 11172 ax-1cn 11173 ax-icn 11174 ax-addcl 11175 ax-addrcl 11176 ax-mulcl 11177 ax-mulrcl 11178 ax-mulcom 11179 ax-addass 11180 ax-mulass 11181 ax-distr 11182 ax-i2m1 11183 ax-1ne0 11184 ax-1rid 11185 ax-rnegex 11186 ax-rrecex 11187 ax-cnre 11188 ax-pre-lttri 11189 ax-pre-lttrn 11190 ax-pre-ltadd 11191 ax-pre-mulgt0 11192 ax-pre-sup 11193 |
| This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-rmo 3371 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4875 df-iun 4960 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7376 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7869 df-1st 7992 df-2nd 7993 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-er 8700 df-en 8950 df-dom 8951 df-sdom 8952 df-sup 9409 df-inf 9410 df-pnf 11260 df-mnf 11261 df-xr 11262 df-ltxr 11263 df-le 11264 df-sub 11458 df-neg 11459 df-div 11887 df-nn 12249 df-n0 12520 df-z 12607 df-uz 12879 df-q 12989 df-ioo 13392 df-ico 13394 df-icc 13395 |
| This theorem is used by: elxrge02 33321 |
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