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Mirrors > Home > MPE Home > Th. List > elii2 | Structured version Visualization version GIF version |
Description: Divide the unit interval into two pieces. (Contributed by Mario Carneiro, 7-Jun-2014.) |
Ref | Expression |
---|---|
elii2 | ⊢ ((𝑋 ∈ (0[,]1) ∧ ¬ 𝑋 ≤ (1 / 2)) → 𝑋 ∈ ((1 / 2)[,]1)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0re 10242 | . . . . 5 ⊢ 0 ∈ ℝ | |
2 | 1re 10241 | . . . . 5 ⊢ 1 ∈ ℝ | |
3 | 1, 2 | elicc2i 12444 | . . . 4 ⊢ (𝑋 ∈ (0[,]1) ↔ (𝑋 ∈ ℝ ∧ 0 ≤ 𝑋 ∧ 𝑋 ≤ 1)) |
4 | 3 | simp1bi 1139 | . . 3 ⊢ (𝑋 ∈ (0[,]1) → 𝑋 ∈ ℝ) |
5 | 4 | adantr 466 | . 2 ⊢ ((𝑋 ∈ (0[,]1) ∧ ¬ 𝑋 ≤ (1 / 2)) → 𝑋 ∈ ℝ) |
6 | halfre 11448 | . . . 4 ⊢ (1 / 2) ∈ ℝ | |
7 | letric 10339 | . . . 4 ⊢ ((𝑋 ∈ ℝ ∧ (1 / 2) ∈ ℝ) → (𝑋 ≤ (1 / 2) ∨ (1 / 2) ≤ 𝑋)) | |
8 | 4, 6, 7 | sylancl 574 | . . 3 ⊢ (𝑋 ∈ (0[,]1) → (𝑋 ≤ (1 / 2) ∨ (1 / 2) ≤ 𝑋)) |
9 | 8 | orcanai 977 | . 2 ⊢ ((𝑋 ∈ (0[,]1) ∧ ¬ 𝑋 ≤ (1 / 2)) → (1 / 2) ≤ 𝑋) |
10 | 3 | simp3bi 1141 | . . 3 ⊢ (𝑋 ∈ (0[,]1) → 𝑋 ≤ 1) |
11 | 10 | adantr 466 | . 2 ⊢ ((𝑋 ∈ (0[,]1) ∧ ¬ 𝑋 ≤ (1 / 2)) → 𝑋 ≤ 1) |
12 | 6, 2 | elicc2i 12444 | . 2 ⊢ (𝑋 ∈ ((1 / 2)[,]1) ↔ (𝑋 ∈ ℝ ∧ (1 / 2) ≤ 𝑋 ∧ 𝑋 ≤ 1)) |
13 | 5, 9, 11, 12 | syl3anbrc 1428 | 1 ⊢ ((𝑋 ∈ (0[,]1) ∧ ¬ 𝑋 ≤ (1 / 2)) → 𝑋 ∈ ((1 / 2)[,]1)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 382 ∨ wo 834 ∈ wcel 2145 class class class wbr 4786 (class class class)co 6793 ℝcr 10137 0cc0 10138 1c1 10139 ≤ cle 10277 / cdiv 10886 2c2 11272 [,]cicc 12383 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7096 ax-cnex 10194 ax-resscn 10195 ax-1cn 10196 ax-icn 10197 ax-addcl 10198 ax-addrcl 10199 ax-mulcl 10200 ax-mulrcl 10201 ax-mulcom 10202 ax-addass 10203 ax-mulass 10204 ax-distr 10205 ax-i2m1 10206 ax-1ne0 10207 ax-1rid 10208 ax-rnegex 10209 ax-rrecex 10210 ax-cnre 10211 ax-pre-lttri 10212 ax-pre-lttrn 10213 ax-pre-ltadd 10214 ax-pre-mulgt0 10215 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 835 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-op 4323 df-uni 4575 df-br 4787 df-opab 4847 df-mpt 4864 df-id 5157 df-po 5170 df-so 5171 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-riota 6754 df-ov 6796 df-oprab 6797 df-mpt2 6798 df-er 7896 df-en 8110 df-dom 8111 df-sdom 8112 df-pnf 10278 df-mnf 10279 df-xr 10280 df-ltxr 10281 df-le 10282 df-sub 10470 df-neg 10471 df-div 10887 df-2 11281 df-icc 12387 |
This theorem is referenced by: phtpycc 23010 copco 23037 pcohtpylem 23038 pcopt 23041 pcopt2 23042 pcoass 23043 pcorevlem 23045 |
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