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Theorem lcvnbtwn3 36638
 Description: The covers relation implies no in-betweenness. (cvnbtwn3 30183 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)
lcvnbtwn3.p (𝜑𝑅𝑈)
lcvnbtwn3.q (𝜑𝑈𝑇)
Assertion
Ref Expression
lcvnbtwn3 (𝜑𝑈 = 𝑅)

Proof of Theorem lcvnbtwn3
StepHypRef Expression
1 lcvnbtwn3.p . 2 (𝜑𝑅𝑈)
2 lcvnbtwn3.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 36635 . . 3 (𝜑 → ¬ (𝑅𝑈𝑈𝑇))
11 iman 405 . . . 4 (((𝑅𝑈𝑈𝑇) → 𝑅 = 𝑈) ↔ ¬ ((𝑅𝑈𝑈𝑇) ∧ ¬ 𝑅 = 𝑈))
12 eqcom 2765 . . . . 5 (𝑈 = 𝑅𝑅 = 𝑈)
1312imbi2i 339 . . . 4 (((𝑅𝑈𝑈𝑇) → 𝑈 = 𝑅) ↔ ((𝑅𝑈𝑈𝑇) → 𝑅 = 𝑈))
14 dfpss2 3993 . . . . . . 7 (𝑅𝑈 ↔ (𝑅𝑈 ∧ ¬ 𝑅 = 𝑈))
1514anbi1i 626 . . . . . 6 ((𝑅𝑈𝑈𝑇) ↔ ((𝑅𝑈 ∧ ¬ 𝑅 = 𝑈) ∧ 𝑈𝑇))
16 an32 645 . . . . . 6 (((𝑅𝑈 ∧ ¬ 𝑅 = 𝑈) ∧ 𝑈𝑇) ↔ ((𝑅𝑈𝑈𝑇) ∧ ¬ 𝑅 = 𝑈))
1715, 16bitri 278 . . . . 5 ((𝑅𝑈𝑈𝑇) ↔ ((𝑅𝑈𝑈𝑇) ∧ ¬ 𝑅 = 𝑈))
1817notbii 323 . . . 4 (¬ (𝑅𝑈𝑈𝑇) ↔ ¬ ((𝑅𝑈𝑈𝑇) ∧ ¬ 𝑅 = 𝑈))
1911, 13, 183bitr4ri 307 . . 3 (¬ (𝑅𝑈𝑈𝑇) ↔ ((𝑅𝑈𝑈𝑇) → 𝑈 = 𝑅))
2010, 19sylib 221 . 2 (𝜑 → ((𝑅𝑈𝑈𝑇) → 𝑈 = 𝑅))
211, 2, 20mp2and 698 1 (𝜑𝑈 = 𝑅)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111   ⊆ wss 3860   ⊊ wpss 3861   class class class wbr 5036  ‘cfv 6340  LSubSpclss 19784   ⋖L clcv 36628 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5173  ax-nul 5180  ax-pow 5238  ax-pr 5302  ax-un 7465 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3699  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-pss 3879  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5037  df-opab 5099  df-mpt 5117  df-id 5434  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-iota 6299  df-fun 6342  df-fv 6348  df-lcv 36629 This theorem is referenced by:  lsatcveq0  36642  lsatcvatlem  36659
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