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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 31270 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 37512 | . . . 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 497 | . 2 ⊢ (𝜑 → ¬ ∃𝑢 ∈ 𝑆 (𝑅 ⊊ 𝑢 ∧ 𝑢 ⊊ 𝑇)) |
10 | lcvnbtwn.u | . . 3 ⊢ (𝜑 → 𝑈 ∈ 𝑆) | |
11 | psseq2 4053 | . . . . 5 ⊢ (𝑢 = 𝑈 → (𝑅 ⊊ 𝑢 ↔ 𝑅 ⊊ 𝑈)) | |
12 | psseq1 4052 | . . . . 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 3584 | . . 3 ⊢ ((𝑈 ∈ 𝑆 ∧ (𝑅 ⊊ 𝑈 ∧ 𝑈 ⊊ 𝑇)) → ∃𝑢 ∈ 𝑆 (𝑅 ⊊ 𝑢 ∧ 𝑢 ⊊ 𝑇)) |
15 | 10, 14 | sylan 581 | . 2 ⊢ ((𝜑 ∧ (𝑅 ⊊ 𝑈 ∧ 𝑈 ⊊ 𝑇)) → ∃𝑢 ∈ 𝑆 (𝑅 ⊊ 𝑢 ∧ 𝑢 ⊊ 𝑇)) |
16 | 9, 15 | mtand 815 | 1 ⊢ (𝜑 → ¬ (𝑅 ⊊ 𝑈 ∧ 𝑈 ⊊ 𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∃wrex 3074 ⊊ wpss 3916 class class class wbr 5110 ‘cfv 6501 LSubSpclss 20408 ⋖L clcv 37509 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-br 5111 df-opab 5173 df-mpt 5194 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-iota 6453 df-fun 6503 df-fv 6509 df-lcv 37510 |
This theorem is referenced by: lcvntr 37517 lcvnbtwn2 37518 lcvnbtwn3 37519 |
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