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Mathbox for Norm Megill |
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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 32048 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 38404 | . . . 4 ⊢ (𝜑 → (𝑅𝐶𝑇 ↔ (𝑅 ⊊ 𝑇 ∧ ¬ ∃𝑢 ∈ 𝑆 (𝑅 ⊊ 𝑢 ∧ 𝑢 ⊊ 𝑇)))) |
8 | 1, 7 | mpbid 231 | . . 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 4083 | . . . . 5 ⊢ (𝑢 = 𝑈 → (𝑅 ⊊ 𝑢 ↔ 𝑅 ⊊ 𝑈)) | |
12 | psseq1 4082 | . . . . 5 ⊢ (𝑢 = 𝑈 → (𝑢 ⊊ 𝑇 ↔ 𝑈 ⊊ 𝑇)) | |
13 | 11, 12 | anbi12d 630 | . . . 4 ⊢ (𝑢 = 𝑈 → ((𝑅 ⊊ 𝑢 ∧ 𝑢 ⊊ 𝑇) ↔ (𝑅 ⊊ 𝑈 ∧ 𝑈 ⊊ 𝑇))) |
14 | 13 | rspcev 3606 | . . 3 ⊢ ((𝑈 ∈ 𝑆 ∧ (𝑅 ⊊ 𝑈 ∧ 𝑈 ⊊ 𝑇)) → ∃𝑢 ∈ 𝑆 (𝑅 ⊊ 𝑢 ∧ 𝑢 ⊊ 𝑇)) |
15 | 10, 14 | sylan 579 | . 2 ⊢ ((𝜑 ∧ (𝑅 ⊊ 𝑈 ∧ 𝑈 ⊊ 𝑇)) → ∃𝑢 ∈ 𝑆 (𝑅 ⊊ 𝑢 ∧ 𝑢 ⊊ 𝑇)) |
16 | 9, 15 | mtand 813 | 1 ⊢ (𝜑 → ¬ (𝑅 ⊊ 𝑈 ∧ 𝑈 ⊊ 𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ∃wrex 3064 ⊊ wpss 3944 class class class wbr 5141 ‘cfv 6537 LSubSpclss 20778 ⋖L clcv 38401 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-iota 6489 df-fun 6539 df-fv 6545 df-lcv 38402 |
This theorem is referenced by: lcvntr 38409 lcvnbtwn2 38410 lcvnbtwn3 38411 |
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