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Theorem lcvnbtwn3 36046
Description: The covers relation implies no in-betweenness. (cvnbtwn3 29993 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 36043 . . 3 (𝜑 → ¬ (𝑅𝑈𝑈𝑇))
11 iman 402 . . . 4 (((𝑅𝑈𝑈𝑇) → 𝑅 = 𝑈) ↔ ¬ ((𝑅𝑈𝑈𝑇) ∧ ¬ 𝑅 = 𝑈))
12 eqcom 2828 . . . . 5 (𝑈 = 𝑅𝑅 = 𝑈)
1312imbi2i 337 . . . 4 (((𝑅𝑈𝑈𝑇) → 𝑈 = 𝑅) ↔ ((𝑅𝑈𝑈𝑇) → 𝑅 = 𝑈))
14 dfpss2 4061 . . . . . . 7 (𝑅𝑈 ↔ (𝑅𝑈 ∧ ¬ 𝑅 = 𝑈))
1514anbi1i 623 . . . . . 6 ((𝑅𝑈𝑈𝑇) ↔ ((𝑅𝑈 ∧ ¬ 𝑅 = 𝑈) ∧ 𝑈𝑇))
16 an32 642 . . . . . 6 (((𝑅𝑈 ∧ ¬ 𝑅 = 𝑈) ∧ 𝑈𝑇) ↔ ((𝑅𝑈𝑈𝑇) ∧ ¬ 𝑅 = 𝑈))
1715, 16bitri 276 . . . . 5 ((𝑅𝑈𝑈𝑇) ↔ ((𝑅𝑈𝑈𝑇) ∧ ¬ 𝑅 = 𝑈))
1817notbii 321 . . . 4 (¬ (𝑅𝑈𝑈𝑇) ↔ ¬ ((𝑅𝑈𝑈𝑇) ∧ ¬ 𝑅 = 𝑈))
1911, 13, 183bitr4ri 305 . . 3 (¬ (𝑅𝑈𝑈𝑇) ↔ ((𝑅𝑈𝑈𝑇) → 𝑈 = 𝑅))
2010, 19sylib 219 . 2 (𝜑 → ((𝑅𝑈𝑈𝑇) → 𝑈 = 𝑅))
211, 2, 20mp2and 695 1 (𝜑𝑈 = 𝑅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396   = wceq 1528  wcel 2105  wss 3935  wpss 3936   class class class wbr 5058  cfv 6349  LSubSpclss 19634  L clcv 36036
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3497  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4833  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-iota 6308  df-fun 6351  df-fv 6357  df-lcv 36037
This theorem is referenced by:  lsatcveq0  36050  lsatcvatlem  36067
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