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Theorem lcvnbtwn2 39028
Description: The covers relation implies no in-betweenness. (cvnbtwn2 32306 analog.) (Contributed by NM, 7-Jan-2015.)
Hypotheses
Ref Expression
lcvnbtwn.s 𝑆 = (LSubSp‘𝑊)
lcvnbtwn.c 𝐶 = ( ⋖L𝑊)
lcvnbtwn.w (𝜑𝑊𝑋)
lcvnbtwn.r (𝜑𝑅𝑆)
lcvnbtwn.t (𝜑𝑇𝑆)
lcvnbtwn.u (𝜑𝑈𝑆)
lcvnbtwn.d (𝜑𝑅𝐶𝑇)
lcvnbtwn2.p (𝜑𝑅𝑈)
lcvnbtwn2.q (𝜑𝑈𝑇)
Assertion
Ref Expression
lcvnbtwn2 (𝜑𝑈 = 𝑇)

Proof of Theorem lcvnbtwn2
StepHypRef Expression
1 lcvnbtwn2.p . 2 (𝜑𝑅𝑈)
2 lcvnbtwn2.q . 2 (𝜑𝑈𝑇)
3 lcvnbtwn.s . . . 4 𝑆 = (LSubSp‘𝑊)
4 lcvnbtwn.c . . . 4 𝐶 = ( ⋖L𝑊)
5 lcvnbtwn.w . . . 4 (𝜑𝑊𝑋)
6 lcvnbtwn.r . . . 4 (𝜑𝑅𝑆)
7 lcvnbtwn.t . . . 4 (𝜑𝑇𝑆)
8 lcvnbtwn.u . . . 4 (𝜑𝑈𝑆)
9 lcvnbtwn.d . . . 4 (𝜑𝑅𝐶𝑇)
103, 4, 5, 6, 7, 8, 9lcvnbtwn 39026 . . 3 (𝜑 → ¬ (𝑅𝑈𝑈𝑇))
11 iman 401 . . . 4 (((𝑅𝑈𝑈𝑇) → 𝑈 = 𝑇) ↔ ¬ ((𝑅𝑈𝑈𝑇) ∧ ¬ 𝑈 = 𝑇))
12 anass 468 . . . . . 6 (((𝑅𝑈𝑈𝑇) ∧ ¬ 𝑈 = 𝑇) ↔ (𝑅𝑈 ∧ (𝑈𝑇 ∧ ¬ 𝑈 = 𝑇)))
13 dfpss2 4088 . . . . . . 7 (𝑈𝑇 ↔ (𝑈𝑇 ∧ ¬ 𝑈 = 𝑇))
1413anbi2i 623 . . . . . 6 ((𝑅𝑈𝑈𝑇) ↔ (𝑅𝑈 ∧ (𝑈𝑇 ∧ ¬ 𝑈 = 𝑇)))
1512, 14bitr4i 278 . . . . 5 (((𝑅𝑈𝑈𝑇) ∧ ¬ 𝑈 = 𝑇) ↔ (𝑅𝑈𝑈𝑇))
1615notbii 320 . . . 4 (¬ ((𝑅𝑈𝑈𝑇) ∧ ¬ 𝑈 = 𝑇) ↔ ¬ (𝑅𝑈𝑈𝑇))
1711, 16bitr2i 276 . . 3 (¬ (𝑅𝑈𝑈𝑇) ↔ ((𝑅𝑈𝑈𝑇) → 𝑈 = 𝑇))
1810, 17sylib 218 . 2 (𝜑 → ((𝑅𝑈𝑈𝑇) → 𝑈 = 𝑇))
191, 2, 18mp2and 699 1 (𝜑𝑈 = 𝑇)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1540  wcel 2108  wss 3951  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:  lcvat  39031  lsatexch  39044
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