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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 12848 | . . . 4 ⊢ (𝑋 ∈ (0[,]1) ↔ (𝑋 ∈ ℝ ∧ 0 ≤ 𝑋 ∧ 𝑋 ≤ 1)) | |
2 | 1 | simp1bi 1141 | . . 3 ⊢ (𝑋 ∈ (0[,]1) → 𝑋 ∈ ℝ) |
3 | 2 | adantr 483 | . 2 ⊢ ((𝑋 ∈ (0[,]1) ∧ ¬ 𝑋 ≤ (1 / 2)) → 𝑋 ∈ ℝ) |
4 | halfre 11845 | . . . 4 ⊢ (1 / 2) ∈ ℝ | |
5 | letric 10734 | . . . 4 ⊢ ((𝑋 ∈ ℝ ∧ (1 / 2) ∈ ℝ) → (𝑋 ≤ (1 / 2) ∨ (1 / 2) ≤ 𝑋)) | |
6 | 2, 4, 5 | sylancl 588 | . . 3 ⊢ (𝑋 ∈ (0[,]1) → (𝑋 ≤ (1 / 2) ∨ (1 / 2) ≤ 𝑋)) |
7 | 6 | orcanai 999 | . 2 ⊢ ((𝑋 ∈ (0[,]1) ∧ ¬ 𝑋 ≤ (1 / 2)) → (1 / 2) ≤ 𝑋) |
8 | 1 | simp3bi 1143 | . . 3 ⊢ (𝑋 ∈ (0[,]1) → 𝑋 ≤ 1) |
9 | 8 | adantr 483 | . 2 ⊢ ((𝑋 ∈ (0[,]1) ∧ ¬ 𝑋 ≤ (1 / 2)) → 𝑋 ≤ 1) |
10 | 1re 10635 | . . 3 ⊢ 1 ∈ ℝ | |
11 | 4, 10 | elicc2i 12796 | . 2 ⊢ (𝑋 ∈ ((1 / 2)[,]1) ↔ (𝑋 ∈ ℝ ∧ (1 / 2) ≤ 𝑋 ∧ 𝑋 ≤ 1)) |
12 | 3, 7, 9, 11 | syl3anbrc 1339 | 1 ⊢ ((𝑋 ∈ (0[,]1) ∧ ¬ 𝑋 ≤ (1 / 2)) → 𝑋 ∈ ((1 / 2)[,]1)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 ∨ wo 843 ∈ wcel 2110 class class class wbr 5058 (class class class)co 7150 ℝcr 10530 0cc0 10531 1c1 10532 ≤ cle 10670 / cdiv 11291 2c2 11686 [,]cicc 12735 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-po 5468 df-so 5469 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-2 11694 df-icc 12739 |
This theorem is referenced by: phtpycc 23589 copco 23616 pcohtpylem 23617 pcopt 23620 pcopt2 23621 pcoass 23622 pcorevlem 23624 |
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