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Theorem lcvnbtwn3 33333
 Description: The covers relation implies no in-betweenness. (cvnbtwn3 28531 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 33330 . . 3 (𝜑 → ¬ (𝑅𝑈𝑈𝑇))
11 iman 439 . . . 4 (((𝑅𝑈𝑈𝑇) → 𝑅 = 𝑈) ↔ ¬ ((𝑅𝑈𝑈𝑇) ∧ ¬ 𝑅 = 𝑈))
12 eqcom 2617 . . . . 5 (𝑈 = 𝑅𝑅 = 𝑈)
1312imbi2i 325 . . . 4 (((𝑅𝑈𝑈𝑇) → 𝑈 = 𝑅) ↔ ((𝑅𝑈𝑈𝑇) → 𝑅 = 𝑈))
14 dfpss2 3654 . . . . . . 7 (𝑅𝑈 ↔ (𝑅𝑈 ∧ ¬ 𝑅 = 𝑈))
1514anbi1i 727 . . . . . 6 ((𝑅𝑈𝑈𝑇) ↔ ((𝑅𝑈 ∧ ¬ 𝑅 = 𝑈) ∧ 𝑈𝑇))
16 an32 835 . . . . . 6 (((𝑅𝑈 ∧ ¬ 𝑅 = 𝑈) ∧ 𝑈𝑇) ↔ ((𝑅𝑈𝑈𝑇) ∧ ¬ 𝑅 = 𝑈))
1715, 16bitri 263 . . . . 5 ((𝑅𝑈𝑈𝑇) ↔ ((𝑅𝑈𝑈𝑇) ∧ ¬ 𝑅 = 𝑈))
1817notbii 309 . . . 4 (¬ (𝑅𝑈𝑈𝑇) ↔ ¬ ((𝑅𝑈𝑈𝑇) ∧ ¬ 𝑅 = 𝑈))
1911, 13, 183bitr4ri 292 . . 3 (¬ (𝑅𝑈𝑈𝑇) ↔ ((𝑅𝑈𝑈𝑇) → 𝑈 = 𝑅))
2010, 19sylib 207 . 2 (𝜑 → ((𝑅𝑈𝑈𝑇) → 𝑈 = 𝑅))
211, 2, 20mp2and 711 1 (𝜑𝑈 = 𝑅)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ⊆ wss 3540   ⊊ wpss 3541   class class class wbr 4583  ‘cfv 5804  LSubSpclss 18753   ⋖L clcv 33323 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-lcv 33324 This theorem is referenced by:  lsatcveq0  33337  lsatcvatlem  33354
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