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| Mirrors > Home > MPE Home > Th. List > Mathboxes > lcvnbtwn3 | Structured version Visualization version GIF version | ||
| Description: The covers relation implies no in-betweenness. (cvnbtwn3 32307 analog.) (Contributed by NM, 7-Jan-2015.) | 
| Ref | Expression | 
|---|---|
| lcvnbtwn.s | ⊢ 𝑆 = (LSubSp‘𝑊) | 
| lcvnbtwn.c | ⊢ 𝐶 = ( ⋖L ‘𝑊) | 
| lcvnbtwn.w | ⊢ (𝜑 → 𝑊 ∈ 𝑋) | 
| lcvnbtwn.r | ⊢ (𝜑 → 𝑅 ∈ 𝑆) | 
| lcvnbtwn.t | ⊢ (𝜑 → 𝑇 ∈ 𝑆) | 
| lcvnbtwn.u | ⊢ (𝜑 → 𝑈 ∈ 𝑆) | 
| lcvnbtwn.d | ⊢ (𝜑 → 𝑅𝐶𝑇) | 
| lcvnbtwn3.p | ⊢ (𝜑 → 𝑅 ⊆ 𝑈) | 
| lcvnbtwn3.q | ⊢ (𝜑 → 𝑈 ⊊ 𝑇) | 
| Ref | Expression | 
|---|---|
| lcvnbtwn3 | ⊢ (𝜑 → 𝑈 = 𝑅) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | lcvnbtwn3.p | . 2 ⊢ (𝜑 → 𝑅 ⊆ 𝑈) | |
| 2 | lcvnbtwn3.q | . 2 ⊢ (𝜑 → 𝑈 ⊊ 𝑇) | |
| 3 | lcvnbtwn.s | . . . 4 ⊢ 𝑆 = (LSubSp‘𝑊) | |
| 4 | lcvnbtwn.c | . . . 4 ⊢ 𝐶 = ( ⋖L ‘𝑊) | |
| 5 | lcvnbtwn.w | . . . 4 ⊢ (𝜑 → 𝑊 ∈ 𝑋) | |
| 6 | lcvnbtwn.r | . . . 4 ⊢ (𝜑 → 𝑅 ∈ 𝑆) | |
| 7 | lcvnbtwn.t | . . . 4 ⊢ (𝜑 → 𝑇 ∈ 𝑆) | |
| 8 | lcvnbtwn.u | . . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝑆) | |
| 9 | lcvnbtwn.d | . . . 4 ⊢ (𝜑 → 𝑅𝐶𝑇) | |
| 10 | 3, 4, 5, 6, 7, 8, 9 | lcvnbtwn 39026 | . . 3 ⊢ (𝜑 → ¬ (𝑅 ⊊ 𝑈 ∧ 𝑈 ⊊ 𝑇)) | 
| 11 | iman 401 | . . . 4 ⊢ (((𝑅 ⊆ 𝑈 ∧ 𝑈 ⊊ 𝑇) → 𝑅 = 𝑈) ↔ ¬ ((𝑅 ⊆ 𝑈 ∧ 𝑈 ⊊ 𝑇) ∧ ¬ 𝑅 = 𝑈)) | |
| 12 | eqcom 2744 | . . . . 5 ⊢ (𝑈 = 𝑅 ↔ 𝑅 = 𝑈) | |
| 13 | 12 | imbi2i 336 | . . . 4 ⊢ (((𝑅 ⊆ 𝑈 ∧ 𝑈 ⊊ 𝑇) → 𝑈 = 𝑅) ↔ ((𝑅 ⊆ 𝑈 ∧ 𝑈 ⊊ 𝑇) → 𝑅 = 𝑈)) | 
| 14 | dfpss2 4088 | . . . . . . 7 ⊢ (𝑅 ⊊ 𝑈 ↔ (𝑅 ⊆ 𝑈 ∧ ¬ 𝑅 = 𝑈)) | |
| 15 | 14 | anbi1i 624 | . . . . . 6 ⊢ ((𝑅 ⊊ 𝑈 ∧ 𝑈 ⊊ 𝑇) ↔ ((𝑅 ⊆ 𝑈 ∧ ¬ 𝑅 = 𝑈) ∧ 𝑈 ⊊ 𝑇)) | 
| 16 | an32 646 | . . . . . 6 ⊢ (((𝑅 ⊆ 𝑈 ∧ ¬ 𝑅 = 𝑈) ∧ 𝑈 ⊊ 𝑇) ↔ ((𝑅 ⊆ 𝑈 ∧ 𝑈 ⊊ 𝑇) ∧ ¬ 𝑅 = 𝑈)) | |
| 17 | 15, 16 | bitri 275 | . . . . 5 ⊢ ((𝑅 ⊊ 𝑈 ∧ 𝑈 ⊊ 𝑇) ↔ ((𝑅 ⊆ 𝑈 ∧ 𝑈 ⊊ 𝑇) ∧ ¬ 𝑅 = 𝑈)) | 
| 18 | 17 | notbii 320 | . . . 4 ⊢ (¬ (𝑅 ⊊ 𝑈 ∧ 𝑈 ⊊ 𝑇) ↔ ¬ ((𝑅 ⊆ 𝑈 ∧ 𝑈 ⊊ 𝑇) ∧ ¬ 𝑅 = 𝑈)) | 
| 19 | 11, 13, 18 | 3bitr4ri 304 | . . 3 ⊢ (¬ (𝑅 ⊊ 𝑈 ∧ 𝑈 ⊊ 𝑇) ↔ ((𝑅 ⊆ 𝑈 ∧ 𝑈 ⊊ 𝑇) → 𝑈 = 𝑅)) | 
| 20 | 10, 19 | sylib 218 | . 2 ⊢ (𝜑 → ((𝑅 ⊆ 𝑈 ∧ 𝑈 ⊊ 𝑇) → 𝑈 = 𝑅)) | 
| 21 | 1, 2, 20 | mp2and 699 | 1 ⊢ (𝜑 → 𝑈 = 𝑅) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ⊆ wss 3951 ⊊ wpss 3952 class class class wbr 5143 ‘cfv 6561 LSubSpclss 20929 ⋖L clcv 39019 | 
| 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 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 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-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 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-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-lcv 39020 | 
| This theorem is referenced by: lsatcveq0 39033 lsatcvatlem 39050 | 
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