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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 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) |
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 36041 | . . 3 ⊢ (𝜑 → ¬ (𝑅 ⊊ 𝑈 ∧ 𝑈 ⊊ 𝑇)) |
11 | iman 402 | . . . 4 ⊢ (((𝑅 ⊆ 𝑈 ∧ 𝑈 ⊊ 𝑇) → 𝑅 = 𝑈) ↔ ¬ ((𝑅 ⊆ 𝑈 ∧ 𝑈 ⊊ 𝑇) ∧ ¬ 𝑅 = 𝑈)) | |
12 | eqcom 2825 | . . . . 5 ⊢ (𝑈 = 𝑅 ↔ 𝑅 = 𝑈) | |
13 | 12 | imbi2i 337 | . . . 4 ⊢ (((𝑅 ⊆ 𝑈 ∧ 𝑈 ⊊ 𝑇) → 𝑈 = 𝑅) ↔ ((𝑅 ⊆ 𝑈 ∧ 𝑈 ⊊ 𝑇) → 𝑅 = 𝑈)) |
14 | dfpss2 4059 | . . . . . . 7 ⊢ (𝑅 ⊊ 𝑈 ↔ (𝑅 ⊆ 𝑈 ∧ ¬ 𝑅 = 𝑈)) | |
15 | 14 | anbi1i 623 | . . . . . 6 ⊢ ((𝑅 ⊊ 𝑈 ∧ 𝑈 ⊊ 𝑇) ↔ ((𝑅 ⊆ 𝑈 ∧ ¬ 𝑅 = 𝑈) ∧ 𝑈 ⊊ 𝑇)) |
16 | an32 642 | . . . . . 6 ⊢ (((𝑅 ⊆ 𝑈 ∧ ¬ 𝑅 = 𝑈) ∧ 𝑈 ⊊ 𝑇) ↔ ((𝑅 ⊆ 𝑈 ∧ 𝑈 ⊊ 𝑇) ∧ ¬ 𝑅 = 𝑈)) | |
17 | 15, 16 | bitri 276 | . . . . 5 ⊢ ((𝑅 ⊊ 𝑈 ∧ 𝑈 ⊊ 𝑇) ↔ ((𝑅 ⊆ 𝑈 ∧ 𝑈 ⊊ 𝑇) ∧ ¬ 𝑅 = 𝑈)) |
18 | 17 | notbii 321 | . . . 4 ⊢ (¬ (𝑅 ⊊ 𝑈 ∧ 𝑈 ⊊ 𝑇) ↔ ¬ ((𝑅 ⊆ 𝑈 ∧ 𝑈 ⊊ 𝑇) ∧ ¬ 𝑅 = 𝑈)) |
19 | 11, 13, 18 | 3bitr4ri 305 | . . 3 ⊢ (¬ (𝑅 ⊊ 𝑈 ∧ 𝑈 ⊊ 𝑇) ↔ ((𝑅 ⊆ 𝑈 ∧ 𝑈 ⊊ 𝑇) → 𝑈 = 𝑅)) |
20 | 10, 19 | sylib 219 | . 2 ⊢ (𝜑 → ((𝑅 ⊆ 𝑈 ∧ 𝑈 ⊊ 𝑇) → 𝑈 = 𝑅)) |
21 | 1, 2, 20 | mp2and 695 | 1 ⊢ (𝜑 → 𝑈 = 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ⊆ wss 3933 ⊊ wpss 3934 class class class wbr 5057 ‘cfv 6348 LSubSpclss 19632 ⋖L clcv 36034 |
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 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 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 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-iota 6307 df-fun 6350 df-fv 6356 df-lcv 36035 |
This theorem is referenced by: lsatcveq0 36048 lsatcvatlem 36065 |
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