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| Mirrors > Home > MPE Home > Th. List > Mathboxes > lcvnbtwn | Structured version Visualization version GIF version | ||
| Description: The covers relation implies no in-betweenness. (cvnbtwn 32305 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) |
| Ref | Expression |
|---|---|
| lcvnbtwn | ⊢ (𝜑 → ¬ (𝑅 ⊊ 𝑈 ∧ 𝑈 ⊊ 𝑇)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | lcvnbtwn.d | . . . 4 ⊢ (𝜑 → 𝑅𝐶𝑇) | |
| 2 | lcvnbtwn.s | . . . . 5 ⊢ 𝑆 = (LSubSp‘𝑊) | |
| 3 | lcvnbtwn.c | . . . . 5 ⊢ 𝐶 = ( ⋖L ‘𝑊) | |
| 4 | lcvnbtwn.w | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ 𝑋) | |
| 5 | lcvnbtwn.r | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ 𝑆) | |
| 6 | lcvnbtwn.t | . . . . 5 ⊢ (𝜑 → 𝑇 ∈ 𝑆) | |
| 7 | 2, 3, 4, 5, 6 | lcvbr 39022 | . . . 4 ⊢ (𝜑 → (𝑅𝐶𝑇 ↔ (𝑅 ⊊ 𝑇 ∧ ¬ ∃𝑢 ∈ 𝑆 (𝑅 ⊊ 𝑢 ∧ 𝑢 ⊊ 𝑇)))) |
| 8 | 1, 7 | mpbid 232 | . . 3 ⊢ (𝜑 → (𝑅 ⊊ 𝑇 ∧ ¬ ∃𝑢 ∈ 𝑆 (𝑅 ⊊ 𝑢 ∧ 𝑢 ⊊ 𝑇))) |
| 9 | 8 | simprd 495 | . 2 ⊢ (𝜑 → ¬ ∃𝑢 ∈ 𝑆 (𝑅 ⊊ 𝑢 ∧ 𝑢 ⊊ 𝑇)) |
| 10 | lcvnbtwn.u | . . 3 ⊢ (𝜑 → 𝑈 ∈ 𝑆) | |
| 11 | psseq2 4091 | . . . . 5 ⊢ (𝑢 = 𝑈 → (𝑅 ⊊ 𝑢 ↔ 𝑅 ⊊ 𝑈)) | |
| 12 | psseq1 4090 | . . . . 5 ⊢ (𝑢 = 𝑈 → (𝑢 ⊊ 𝑇 ↔ 𝑈 ⊊ 𝑇)) | |
| 13 | 11, 12 | anbi12d 632 | . . . 4 ⊢ (𝑢 = 𝑈 → ((𝑅 ⊊ 𝑢 ∧ 𝑢 ⊊ 𝑇) ↔ (𝑅 ⊊ 𝑈 ∧ 𝑈 ⊊ 𝑇))) |
| 14 | 13 | rspcev 3622 | . . 3 ⊢ ((𝑈 ∈ 𝑆 ∧ (𝑅 ⊊ 𝑈 ∧ 𝑈 ⊊ 𝑇)) → ∃𝑢 ∈ 𝑆 (𝑅 ⊊ 𝑢 ∧ 𝑢 ⊊ 𝑇)) |
| 15 | 10, 14 | sylan 580 | . 2 ⊢ ((𝜑 ∧ (𝑅 ⊊ 𝑈 ∧ 𝑈 ⊊ 𝑇)) → ∃𝑢 ∈ 𝑆 (𝑅 ⊊ 𝑢 ∧ 𝑢 ⊊ 𝑇)) |
| 16 | 9, 15 | mtand 816 | 1 ⊢ (𝜑 → ¬ (𝑅 ⊊ 𝑈 ∧ 𝑈 ⊊ 𝑇)) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∃wrex 3070 ⊊ 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: lcvntr 39027 lcvnbtwn2 39028 lcvnbtwn3 39029 |
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