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Theorem lcvnbtwn2 36285
Description: The covers relation implies no in-betweenness. (cvnbtwn2 30068 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 36283 . . 3 (𝜑 → ¬ (𝑅𝑈𝑈𝑇))
11 iman 405 . . . 4 (((𝑅𝑈𝑈𝑇) → 𝑈 = 𝑇) ↔ ¬ ((𝑅𝑈𝑈𝑇) ∧ ¬ 𝑈 = 𝑇))
12 anass 472 . . . . . 6 (((𝑅𝑈𝑈𝑇) ∧ ¬ 𝑈 = 𝑇) ↔ (𝑅𝑈 ∧ (𝑈𝑇 ∧ ¬ 𝑈 = 𝑇)))
13 dfpss2 4037 . . . . . . 7 (𝑈𝑇 ↔ (𝑈𝑇 ∧ ¬ 𝑈 = 𝑇))
1413anbi2i 625 . . . . . 6 ((𝑅𝑈𝑈𝑇) ↔ (𝑅𝑈 ∧ (𝑈𝑇 ∧ ¬ 𝑈 = 𝑇)))
1512, 14bitr4i 281 . . . . 5 (((𝑅𝑈𝑈𝑇) ∧ ¬ 𝑈 = 𝑇) ↔ (𝑅𝑈𝑈𝑇))
1615notbii 323 . . . 4 (¬ ((𝑅𝑈𝑈𝑇) ∧ ¬ 𝑈 = 𝑇) ↔ ¬ (𝑅𝑈𝑈𝑇))
1711, 16bitr2i 279 . . 3 (¬ (𝑅𝑈𝑈𝑇) ↔ ((𝑅𝑈𝑈𝑇) → 𝑈 = 𝑇))
1810, 17sylib 221 . 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 399   = wceq 1538  wcel 2114  wss 3908  wpss 3909   class class class wbr 5042  cfv 6334  LSubSpclss 19694  L clcv 36276
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-iota 6293  df-fun 6336  df-fv 6342  df-lcv 36277
This theorem is referenced by:  lcvat  36288  lsatexch  36301
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