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Theorem lcvnbtwn2 38555
Description: The covers relation implies no in-betweenness. (cvnbtwn2 32141 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 38553 . . 3 (𝜑 → ¬ (𝑅𝑈𝑈𝑇))
11 iman 400 . . . 4 (((𝑅𝑈𝑈𝑇) → 𝑈 = 𝑇) ↔ ¬ ((𝑅𝑈𝑈𝑇) ∧ ¬ 𝑈 = 𝑇))
12 anass 467 . . . . . 6 (((𝑅𝑈𝑈𝑇) ∧ ¬ 𝑈 = 𝑇) ↔ (𝑅𝑈 ∧ (𝑈𝑇 ∧ ¬ 𝑈 = 𝑇)))
13 dfpss2 4077 . . . . . . 7 (𝑈𝑇 ↔ (𝑈𝑇 ∧ ¬ 𝑈 = 𝑇))
1413anbi2i 621 . . . . . 6 ((𝑅𝑈𝑈𝑇) ↔ (𝑅𝑈 ∧ (𝑈𝑇 ∧ ¬ 𝑈 = 𝑇)))
1512, 14bitr4i 277 . . . . 5 (((𝑅𝑈𝑈𝑇) ∧ ¬ 𝑈 = 𝑇) ↔ (𝑅𝑈𝑈𝑇))
1615notbii 319 . . . 4 (¬ ((𝑅𝑈𝑈𝑇) ∧ ¬ 𝑈 = 𝑇) ↔ ¬ (𝑅𝑈𝑈𝑇))
1711, 16bitr2i 275 . . 3 (¬ (𝑅𝑈𝑈𝑇) ↔ ((𝑅𝑈𝑈𝑇) → 𝑈 = 𝑇))
1810, 17sylib 217 . 2 (𝜑 → ((𝑅𝑈𝑈𝑇) → 𝑈 = 𝑇))
191, 2, 18mp2and 697 1 (𝜑𝑈 = 𝑇)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 394   = wceq 1533  wcel 2098  wss 3939  wpss 3940   class class class wbr 5143  cfv 6543  LSubSpclss 20819  L clcv 38546
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7738
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-pss 3959  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-iota 6495  df-fun 6545  df-fv 6551  df-lcv 38547
This theorem is referenced by:  lcvat  38558  lsatexch  38571
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