Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  lcvnbtwn3 Structured version   Visualization version   GIF version

Theorem lcvnbtwn3 34787
Description: The covers relation implies no in-betweenness. (cvnbtwn3 29427 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 34784 . . 3 (𝜑 → ¬ (𝑅𝑈𝑈𝑇))
11 iman 439 . . . 4 (((𝑅𝑈𝑈𝑇) → 𝑅 = 𝑈) ↔ ¬ ((𝑅𝑈𝑈𝑇) ∧ ¬ 𝑅 = 𝑈))
12 eqcom 2755 . . . . 5 (𝑈 = 𝑅𝑅 = 𝑈)
1312imbi2i 325 . . . 4 (((𝑅𝑈𝑈𝑇) → 𝑈 = 𝑅) ↔ ((𝑅𝑈𝑈𝑇) → 𝑅 = 𝑈))
14 dfpss2 3822 . . . . . . 7 (𝑅𝑈 ↔ (𝑅𝑈 ∧ ¬ 𝑅 = 𝑈))
1514anbi1i 733 . . . . . 6 ((𝑅𝑈𝑈𝑇) ↔ ((𝑅𝑈 ∧ ¬ 𝑅 = 𝑈) ∧ 𝑈𝑇))
16 an32 874 . . . . . 6 (((𝑅𝑈 ∧ ¬ 𝑅 = 𝑈) ∧ 𝑈𝑇) ↔ ((𝑅𝑈𝑈𝑇) ∧ ¬ 𝑅 = 𝑈))
1715, 16bitri 264 . . . . 5 ((𝑅𝑈𝑈𝑇) ↔ ((𝑅𝑈𝑈𝑇) ∧ ¬ 𝑅 = 𝑈))
1817notbii 309 . . . 4 (¬ (𝑅𝑈𝑈𝑇) ↔ ¬ ((𝑅𝑈𝑈𝑇) ∧ ¬ 𝑅 = 𝑈))
1911, 13, 183bitr4ri 293 . . 3 (¬ (𝑅𝑈𝑈𝑇) ↔ ((𝑅𝑈𝑈𝑇) → 𝑈 = 𝑅))
2010, 19sylib 208 . 2 (𝜑 → ((𝑅𝑈𝑈𝑇) → 𝑈 = 𝑅))
211, 2, 20mp2and 717 1 (𝜑𝑈 = 𝑅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383   = wceq 1620  wcel 2127  wss 3703  wpss 3704   class class class wbr 4792  cfv 6037  LSubSpclss 19105  L clcv 34777
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-8 2129  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-sep 4921  ax-nul 4929  ax-pow 4980  ax-pr 5043  ax-un 7102
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ne 2921  df-ral 3043  df-rex 3044  df-rab 3047  df-v 3330  df-sbc 3565  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-pss 3719  df-nul 4047  df-if 4219  df-pw 4292  df-sn 4310  df-pr 4312  df-op 4316  df-uni 4577  df-br 4793  df-opab 4853  df-mpt 4870  df-id 5162  df-xp 5260  df-rel 5261  df-cnv 5262  df-co 5263  df-dm 5264  df-iota 6000  df-fun 6039  df-fv 6045  df-lcv 34778
This theorem is referenced by:  lsatcveq0  34791  lsatcvatlem  34808
  Copyright terms: Public domain W3C validator