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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 29992 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 36043 | . . 3 ⊢ (𝜑 → ¬ (𝑅 ⊊ 𝑈 ∧ 𝑈 ⊊ 𝑇)) |
11 | iman 402 | . . . 4 ⊢ (((𝑅 ⊊ 𝑈 ∧ 𝑈 ⊆ 𝑇) → 𝑈 = 𝑇) ↔ ¬ ((𝑅 ⊊ 𝑈 ∧ 𝑈 ⊆ 𝑇) ∧ ¬ 𝑈 = 𝑇)) | |
12 | anass 469 | . . . . . 6 ⊢ (((𝑅 ⊊ 𝑈 ∧ 𝑈 ⊆ 𝑇) ∧ ¬ 𝑈 = 𝑇) ↔ (𝑅 ⊊ 𝑈 ∧ (𝑈 ⊆ 𝑇 ∧ ¬ 𝑈 = 𝑇))) | |
13 | dfpss2 4061 | . . . . . . 7 ⊢ (𝑈 ⊊ 𝑇 ↔ (𝑈 ⊆ 𝑇 ∧ ¬ 𝑈 = 𝑇)) | |
14 | 13 | anbi2i 622 | . . . . . 6 ⊢ ((𝑅 ⊊ 𝑈 ∧ 𝑈 ⊊ 𝑇) ↔ (𝑅 ⊊ 𝑈 ∧ (𝑈 ⊆ 𝑇 ∧ ¬ 𝑈 = 𝑇))) |
15 | 12, 14 | bitr4i 279 | . . . . 5 ⊢ (((𝑅 ⊊ 𝑈 ∧ 𝑈 ⊆ 𝑇) ∧ ¬ 𝑈 = 𝑇) ↔ (𝑅 ⊊ 𝑈 ∧ 𝑈 ⊊ 𝑇)) |
16 | 15 | notbii 321 | . . . 4 ⊢ (¬ ((𝑅 ⊊ 𝑈 ∧ 𝑈 ⊆ 𝑇) ∧ ¬ 𝑈 = 𝑇) ↔ ¬ (𝑅 ⊊ 𝑈 ∧ 𝑈 ⊊ 𝑇)) |
17 | 11, 16 | bitr2i 277 | . . 3 ⊢ (¬ (𝑅 ⊊ 𝑈 ∧ 𝑈 ⊊ 𝑇) ↔ ((𝑅 ⊊ 𝑈 ∧ 𝑈 ⊆ 𝑇) → 𝑈 = 𝑇)) |
18 | 10, 17 | sylib 219 | . 2 ⊢ (𝜑 → ((𝑅 ⊊ 𝑈 ∧ 𝑈 ⊆ 𝑇) → 𝑈 = 𝑇)) |
19 | 1, 2, 18 | mp2and 695 | 1 ⊢ (𝜑 → 𝑈 = 𝑇) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ⊆ wss 3935 ⊊ wpss 3936 class class class wbr 5058 ‘cfv 6349 LSubSpclss 19634 ⋖L clcv 36036 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4833 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-iota 6308 df-fun 6351 df-fv 6357 df-lcv 36037 |
This theorem is referenced by: lcvat 36048 lsatexch 36061 |
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