Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  lcvnbtwn2 Structured version   Visualization version   GIF version

Theorem lcvnbtwn2 37518
Description: The covers relation implies no in-betweenness. (cvnbtwn2 31271 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 37516 . . 3 (𝜑 → ¬ (𝑅𝑈𝑈𝑇))
11 iman 403 . . . 4 (((𝑅𝑈𝑈𝑇) → 𝑈 = 𝑇) ↔ ¬ ((𝑅𝑈𝑈𝑇) ∧ ¬ 𝑈 = 𝑇))
12 anass 470 . . . . . 6 (((𝑅𝑈𝑈𝑇) ∧ ¬ 𝑈 = 𝑇) ↔ (𝑅𝑈 ∧ (𝑈𝑇 ∧ ¬ 𝑈 = 𝑇)))
13 dfpss2 4050 . . . . . . 7 (𝑈𝑇 ↔ (𝑈𝑇 ∧ ¬ 𝑈 = 𝑇))
1413anbi2i 624 . . . . . 6 ((𝑅𝑈𝑈𝑇) ↔ (𝑅𝑈 ∧ (𝑈𝑇 ∧ ¬ 𝑈 = 𝑇)))
1512, 14bitr4i 278 . . . . 5 (((𝑅𝑈𝑈𝑇) ∧ ¬ 𝑈 = 𝑇) ↔ (𝑅𝑈𝑈𝑇))
1615notbii 320 . . . 4 (¬ ((𝑅𝑈𝑈𝑇) ∧ ¬ 𝑈 = 𝑇) ↔ ¬ (𝑅𝑈𝑈𝑇))
1711, 16bitr2i 276 . . 3 (¬ (𝑅𝑈𝑈𝑇) ↔ ((𝑅𝑈𝑈𝑇) → 𝑈 = 𝑇))
1810, 17sylib 217 . 2 (𝜑 → ((𝑅𝑈𝑈𝑇) → 𝑈 = 𝑇))
191, 2, 18mp2and 698 1 (𝜑𝑈 = 𝑇)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 397   = wceq 1542  wcel 2107  wss 3915  wpss 3916   class class class wbr 5110  cfv 6501  LSubSpclss 20408  L clcv 37509
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-iota 6453  df-fun 6503  df-fv 6509  df-lcv 37510
This theorem is referenced by:  lcvat  37521  lsatexch  37534
  Copyright terms: Public domain W3C validator