| Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > Mathboxes > iccdifprioo | Structured version Visualization version GIF version | ||
| Description: An open interval is the closed interval without the bounds. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
| Ref | Expression |
|---|---|
| iccdifprioo | ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → ((𝐴[,]𝐵) ∖ {𝐴, 𝐵}) = (𝐴(,)𝐵)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | prunioo 13521 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) → ((𝐴(,)𝐵) ∪ {𝐴, 𝐵}) = (𝐴[,]𝐵)) | |
| 2 | 1 | eqcomd 2743 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) → (𝐴[,]𝐵) = ((𝐴(,)𝐵) ∪ {𝐴, 𝐵})) |
| 3 | 2 | difeq1d 4125 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) → ((𝐴[,]𝐵) ∖ {𝐴, 𝐵}) = (((𝐴(,)𝐵) ∪ {𝐴, 𝐵}) ∖ {𝐴, 𝐵})) |
| 4 | difun2 4481 | . . . . 5 ⊢ (((𝐴(,)𝐵) ∪ {𝐴, 𝐵}) ∖ {𝐴, 𝐵}) = ((𝐴(,)𝐵) ∖ {𝐴, 𝐵}) | |
| 5 | iooinlbub 45514 | . . . . . 6 ⊢ ((𝐴(,)𝐵) ∩ {𝐴, 𝐵}) = ∅ | |
| 6 | disj3 4454 | . . . . . 6 ⊢ (((𝐴(,)𝐵) ∩ {𝐴, 𝐵}) = ∅ ↔ (𝐴(,)𝐵) = ((𝐴(,)𝐵) ∖ {𝐴, 𝐵})) | |
| 7 | 5, 6 | mpbi 230 | . . . . 5 ⊢ (𝐴(,)𝐵) = ((𝐴(,)𝐵) ∖ {𝐴, 𝐵}) |
| 8 | 4, 7 | eqtr4i 2768 | . . . 4 ⊢ (((𝐴(,)𝐵) ∪ {𝐴, 𝐵}) ∖ {𝐴, 𝐵}) = (𝐴(,)𝐵) |
| 9 | 3, 8 | eqtrdi 2793 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵) → ((𝐴[,]𝐵) ∖ {𝐴, 𝐵}) = (𝐴(,)𝐵)) |
| 10 | 9 | 3expa 1119 | . 2 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ 𝐴 ≤ 𝐵) → ((𝐴[,]𝐵) ∖ {𝐴, 𝐵}) = (𝐴(,)𝐵)) |
| 11 | difssd 4137 | . . . . 5 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ ¬ 𝐴 ≤ 𝐵) → ((𝐴[,]𝐵) ∖ {𝐴, 𝐵}) ⊆ (𝐴[,]𝐵)) | |
| 12 | simpr 484 | . . . . . . . 8 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ ¬ 𝐴 ≤ 𝐵) → ¬ 𝐴 ≤ 𝐵) | |
| 13 | xrlenlt 11326 | . . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴)) | |
| 14 | 13 | adantr 480 | . . . . . . . 8 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ ¬ 𝐴 ≤ 𝐵) → (𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴)) |
| 15 | 12, 14 | mtbid 324 | . . . . . . 7 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ ¬ 𝐴 ≤ 𝐵) → ¬ ¬ 𝐵 < 𝐴) |
| 16 | 15 | notnotrd 133 | . . . . . 6 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ ¬ 𝐴 ≤ 𝐵) → 𝐵 < 𝐴) |
| 17 | icc0 13435 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → ((𝐴[,]𝐵) = ∅ ↔ 𝐵 < 𝐴)) | |
| 18 | 17 | adantr 480 | . . . . . 6 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ ¬ 𝐴 ≤ 𝐵) → ((𝐴[,]𝐵) = ∅ ↔ 𝐵 < 𝐴)) |
| 19 | 16, 18 | mpbird 257 | . . . . 5 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ ¬ 𝐴 ≤ 𝐵) → (𝐴[,]𝐵) = ∅) |
| 20 | 11, 19 | sseqtrd 4020 | . . . 4 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ ¬ 𝐴 ≤ 𝐵) → ((𝐴[,]𝐵) ∖ {𝐴, 𝐵}) ⊆ ∅) |
| 21 | ss0 4402 | . . . 4 ⊢ (((𝐴[,]𝐵) ∖ {𝐴, 𝐵}) ⊆ ∅ → ((𝐴[,]𝐵) ∖ {𝐴, 𝐵}) = ∅) | |
| 22 | 20, 21 | syl 17 | . . 3 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ ¬ 𝐴 ≤ 𝐵) → ((𝐴[,]𝐵) ∖ {𝐴, 𝐵}) = ∅) |
| 23 | simplr 769 | . . . . 5 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ ¬ 𝐴 ≤ 𝐵) → 𝐵 ∈ ℝ*) | |
| 24 | simpll 767 | . . . . 5 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ ¬ 𝐴 ≤ 𝐵) → 𝐴 ∈ ℝ*) | |
| 25 | 23, 24, 16 | xrltled 13192 | . . . 4 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ ¬ 𝐴 ≤ 𝐵) → 𝐵 ≤ 𝐴) |
| 26 | ioo0 13412 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → ((𝐴(,)𝐵) = ∅ ↔ 𝐵 ≤ 𝐴)) | |
| 27 | 26 | adantr 480 | . . . 4 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ ¬ 𝐴 ≤ 𝐵) → ((𝐴(,)𝐵) = ∅ ↔ 𝐵 ≤ 𝐴)) |
| 28 | 25, 27 | mpbird 257 | . . 3 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ ¬ 𝐴 ≤ 𝐵) → (𝐴(,)𝐵) = ∅) |
| 29 | 22, 28 | eqtr4d 2780 | . 2 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) ∧ ¬ 𝐴 ≤ 𝐵) → ((𝐴[,]𝐵) ∖ {𝐴, 𝐵}) = (𝐴(,)𝐵)) |
| 30 | 10, 29 | pm2.61dan 813 | 1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → ((𝐴[,]𝐵) ∖ {𝐴, 𝐵}) = (𝐴(,)𝐵)) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ∖ cdif 3948 ∪ cun 3949 ∩ cin 3950 ⊆ wss 3951 ∅c0 4333 {cpr 4628 class class class wbr 5143 (class class class)co 7431 ℝ*cxr 11294 < clt 11295 ≤ cle 11296 (,)cioo 13387 [,]cicc 13390 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-sup 9482 df-inf 9483 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-n0 12527 df-z 12614 df-uz 12879 df-q 12991 df-ioo 13391 df-ico 13393 df-icc 13394 |
| This theorem is referenced by: (None) |
| Copyright terms: Public domain | W3C validator |