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| Mirrors > Home > MPE Home > Th. List > Mathboxes > lcvnbtwn2 | Structured version Visualization version GIF version | ||
| Description: The covers relation implies no in-betweenness. (cvnbtwn2 32306 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) |
| lcvnbtwn2.p | ⊢ (𝜑 → 𝑅 ⊊ 𝑈) |
| lcvnbtwn2.q | ⊢ (𝜑 → 𝑈 ⊆ 𝑇) |
| Ref | Expression |
|---|---|
| lcvnbtwn2 | ⊢ (𝜑 → 𝑈 = 𝑇) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | lcvnbtwn2.p | . 2 ⊢ (𝜑 → 𝑅 ⊊ 𝑈) | |
| 2 | lcvnbtwn2.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) → 𝑈 = 𝑇) ↔ ¬ ((𝑅 ⊊ 𝑈 ∧ 𝑈 ⊆ 𝑇) ∧ ¬ 𝑈 = 𝑇)) | |
| 12 | anass 468 | . . . . . 6 ⊢ (((𝑅 ⊊ 𝑈 ∧ 𝑈 ⊆ 𝑇) ∧ ¬ 𝑈 = 𝑇) ↔ (𝑅 ⊊ 𝑈 ∧ (𝑈 ⊆ 𝑇 ∧ ¬ 𝑈 = 𝑇))) | |
| 13 | dfpss2 4088 | . . . . . . 7 ⊢ (𝑈 ⊊ 𝑇 ↔ (𝑈 ⊆ 𝑇 ∧ ¬ 𝑈 = 𝑇)) | |
| 14 | 13 | anbi2i 623 | . . . . . 6 ⊢ ((𝑅 ⊊ 𝑈 ∧ 𝑈 ⊊ 𝑇) ↔ (𝑅 ⊊ 𝑈 ∧ (𝑈 ⊆ 𝑇 ∧ ¬ 𝑈 = 𝑇))) |
| 15 | 12, 14 | bitr4i 278 | . . . . 5 ⊢ (((𝑅 ⊊ 𝑈 ∧ 𝑈 ⊆ 𝑇) ∧ ¬ 𝑈 = 𝑇) ↔ (𝑅 ⊊ 𝑈 ∧ 𝑈 ⊊ 𝑇)) |
| 16 | 15 | notbii 320 | . . . 4 ⊢ (¬ ((𝑅 ⊊ 𝑈 ∧ 𝑈 ⊆ 𝑇) ∧ ¬ 𝑈 = 𝑇) ↔ ¬ (𝑅 ⊊ 𝑈 ∧ 𝑈 ⊊ 𝑇)) |
| 17 | 11, 16 | bitr2i 276 | . . 3 ⊢ (¬ (𝑅 ⊊ 𝑈 ∧ 𝑈 ⊊ 𝑇) ↔ ((𝑅 ⊊ 𝑈 ∧ 𝑈 ⊆ 𝑇) → 𝑈 = 𝑇)) |
| 18 | 10, 17 | sylib 218 | . 2 ⊢ (𝜑 → ((𝑅 ⊊ 𝑈 ∧ 𝑈 ⊆ 𝑇) → 𝑈 = 𝑇)) |
| 19 | 1, 2, 18 | mp2and 699 | 1 ⊢ (𝜑 → 𝑈 = 𝑇) |
| 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: lcvat 39031 lsatexch 39044 |
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