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Mirrors > Home > MPE Home > Th. List > elii1 | Structured version Visualization version GIF version |
Description: Divide the unit interval into two pieces. (Contributed by Mario Carneiro, 7-Jun-2014.) |
Ref | Expression |
---|---|
elii1 | ⊢ (𝑋 ∈ (0[,](1 / 2)) ↔ (𝑋 ∈ (0[,]1) ∧ 𝑋 ≤ (1 / 2))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0re 10632 | . . . . . 6 ⊢ 0 ∈ ℝ | |
2 | halfre 11839 | . . . . . 6 ⊢ (1 / 2) ∈ ℝ | |
3 | 1, 2 | elicc2i 12791 | . . . . 5 ⊢ (𝑋 ∈ (0[,](1 / 2)) ↔ (𝑋 ∈ ℝ ∧ 0 ≤ 𝑋 ∧ 𝑋 ≤ (1 / 2))) |
4 | 3 | simp1bi 1142 | . . . 4 ⊢ (𝑋 ∈ (0[,](1 / 2)) → 𝑋 ∈ ℝ) |
5 | 2 | a1i 11 | . . . 4 ⊢ (𝑋 ∈ (0[,](1 / 2)) → (1 / 2) ∈ ℝ) |
6 | 1re 10630 | . . . . 5 ⊢ 1 ∈ ℝ | |
7 | 6 | a1i 11 | . . . 4 ⊢ (𝑋 ∈ (0[,](1 / 2)) → 1 ∈ ℝ) |
8 | 3 | simp3bi 1144 | . . . 4 ⊢ (𝑋 ∈ (0[,](1 / 2)) → 𝑋 ≤ (1 / 2)) |
9 | halflt1 11843 | . . . . . 6 ⊢ (1 / 2) < 1 | |
10 | 2, 6, 9 | ltleii 10752 | . . . . 5 ⊢ (1 / 2) ≤ 1 |
11 | 10 | a1i 11 | . . . 4 ⊢ (𝑋 ∈ (0[,](1 / 2)) → (1 / 2) ≤ 1) |
12 | 4, 5, 7, 8, 11 | letrd 10786 | . . 3 ⊢ (𝑋 ∈ (0[,](1 / 2)) → 𝑋 ≤ 1) |
13 | 12 | pm4.71ri 564 | . 2 ⊢ (𝑋 ∈ (0[,](1 / 2)) ↔ (𝑋 ≤ 1 ∧ 𝑋 ∈ (0[,](1 / 2)))) |
14 | ancom 464 | . . 3 ⊢ ((𝑋 ≤ 1 ∧ 𝑋 ∈ (0[,](1 / 2))) ↔ (𝑋 ∈ (0[,](1 / 2)) ∧ 𝑋 ≤ 1)) | |
15 | an32 645 | . . . 4 ⊢ ((((𝑋 ∈ ℝ ∧ 0 ≤ 𝑋) ∧ 𝑋 ≤ (1 / 2)) ∧ 𝑋 ≤ 1) ↔ (((𝑋 ∈ ℝ ∧ 0 ≤ 𝑋) ∧ 𝑋 ≤ 1) ∧ 𝑋 ≤ (1 / 2))) | |
16 | df-3an 1086 | . . . . . 6 ⊢ ((𝑋 ∈ ℝ ∧ 0 ≤ 𝑋 ∧ 𝑋 ≤ (1 / 2)) ↔ ((𝑋 ∈ ℝ ∧ 0 ≤ 𝑋) ∧ 𝑋 ≤ (1 / 2))) | |
17 | 3, 16 | bitri 278 | . . . . 5 ⊢ (𝑋 ∈ (0[,](1 / 2)) ↔ ((𝑋 ∈ ℝ ∧ 0 ≤ 𝑋) ∧ 𝑋 ≤ (1 / 2))) |
18 | 17 | anbi1i 626 | . . . 4 ⊢ ((𝑋 ∈ (0[,](1 / 2)) ∧ 𝑋 ≤ 1) ↔ (((𝑋 ∈ ℝ ∧ 0 ≤ 𝑋) ∧ 𝑋 ≤ (1 / 2)) ∧ 𝑋 ≤ 1)) |
19 | 1, 6 | elicc2i 12791 | . . . . . 6 ⊢ (𝑋 ∈ (0[,]1) ↔ (𝑋 ∈ ℝ ∧ 0 ≤ 𝑋 ∧ 𝑋 ≤ 1)) |
20 | df-3an 1086 | . . . . . 6 ⊢ ((𝑋 ∈ ℝ ∧ 0 ≤ 𝑋 ∧ 𝑋 ≤ 1) ↔ ((𝑋 ∈ ℝ ∧ 0 ≤ 𝑋) ∧ 𝑋 ≤ 1)) | |
21 | 19, 20 | bitri 278 | . . . . 5 ⊢ (𝑋 ∈ (0[,]1) ↔ ((𝑋 ∈ ℝ ∧ 0 ≤ 𝑋) ∧ 𝑋 ≤ 1)) |
22 | 21 | anbi1i 626 | . . . 4 ⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑋 ≤ (1 / 2)) ↔ (((𝑋 ∈ ℝ ∧ 0 ≤ 𝑋) ∧ 𝑋 ≤ 1) ∧ 𝑋 ≤ (1 / 2))) |
23 | 15, 18, 22 | 3bitr4i 306 | . . 3 ⊢ ((𝑋 ∈ (0[,](1 / 2)) ∧ 𝑋 ≤ 1) ↔ (𝑋 ∈ (0[,]1) ∧ 𝑋 ≤ (1 / 2))) |
24 | 14, 23 | bitri 278 | . 2 ⊢ ((𝑋 ≤ 1 ∧ 𝑋 ∈ (0[,](1 / 2))) ↔ (𝑋 ∈ (0[,]1) ∧ 𝑋 ≤ (1 / 2))) |
25 | 13, 24 | bitri 278 | 1 ⊢ (𝑋 ∈ (0[,](1 / 2)) ↔ (𝑋 ∈ (0[,]1) ∧ 𝑋 ≤ (1 / 2))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 ∧ w3a 1084 ∈ wcel 2111 class class class wbr 5030 (class class class)co 7135 ℝcr 10525 0cc0 10526 1c1 10527 ≤ cle 10665 / cdiv 11286 2c2 11680 [,]cicc 12729 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-po 5438 df-so 5439 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-2 11688 df-icc 12733 |
This theorem is referenced by: phtpycc 23596 pcoval1 23618 copco 23623 pcohtpylem 23624 pcopt 23627 pcopt2 23628 pcorevlem 23631 |
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