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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 | elicc01 13450 | . . . 4 ⊢ (𝑋 ∈ (0[,]1) ↔ (𝑋 ∈ ℝ ∧ 0 ≤ 𝑋 ∧ 𝑋 ≤ 1)) | |
2 | 1 | simp1bi 1144 | . . 3 ⊢ (𝑋 ∈ (0[,]1) → 𝑋 ∈ ℝ) |
3 | 2 | adantr 480 | . 2 ⊢ ((𝑋 ∈ (0[,]1) ∧ ¬ 𝑋 ≤ (1 / 2)) → 𝑋 ∈ ℝ) |
4 | halfre 12433 | . . . 4 ⊢ (1 / 2) ∈ ℝ | |
5 | letric 11321 | . . . 4 ⊢ ((𝑋 ∈ ℝ ∧ (1 / 2) ∈ ℝ) → (𝑋 ≤ (1 / 2) ∨ (1 / 2) ≤ 𝑋)) | |
6 | 2, 4, 5 | sylancl 585 | . . 3 ⊢ (𝑋 ∈ (0[,]1) → (𝑋 ≤ (1 / 2) ∨ (1 / 2) ≤ 𝑋)) |
7 | 6 | orcanai 1000 | . 2 ⊢ ((𝑋 ∈ (0[,]1) ∧ ¬ 𝑋 ≤ (1 / 2)) → (1 / 2) ≤ 𝑋) |
8 | 1 | simp3bi 1146 | . . 3 ⊢ (𝑋 ∈ (0[,]1) → 𝑋 ≤ 1) |
9 | 8 | adantr 480 | . 2 ⊢ ((𝑋 ∈ (0[,]1) ∧ ¬ 𝑋 ≤ (1 / 2)) → 𝑋 ≤ 1) |
10 | 1re 11221 | . . 3 ⊢ 1 ∈ ℝ | |
11 | 4, 10 | elicc2i 13397 | . 2 ⊢ (𝑋 ∈ ((1 / 2)[,]1) ↔ (𝑋 ∈ ℝ ∧ (1 / 2) ≤ 𝑋 ∧ 𝑋 ≤ 1)) |
12 | 3, 7, 9, 11 | syl3anbrc 1342 | 1 ⊢ ((𝑋 ∈ (0[,]1) ∧ ¬ 𝑋 ≤ (1 / 2)) → 𝑋 ∈ ((1 / 2)[,]1)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∨ wo 844 ∈ wcel 2105 class class class wbr 5148 (class class class)co 7412 ℝcr 11115 0cc0 11116 1c1 11117 ≤ cle 11256 / cdiv 11878 2c2 12274 [,]cicc 13334 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-po 5588 df-so 5589 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-er 8709 df-en 8946 df-dom 8947 df-sdom 8948 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-div 11879 df-2 12282 df-icc 13338 |
This theorem is referenced by: phtpycc 24750 copco 24778 pcohtpylem 24779 pcopt 24782 pcopt2 24783 pcoass 24784 pcorevlem 24786 |
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