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Theorem lcvnbtwn2 33833
Description: The covers relation implies no in-betweenness. (cvnbtwn2 29034 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 33831 . . 3 (𝜑 → ¬ (𝑅𝑈𝑈𝑇))
11 iman 440 . . . 4 (((𝑅𝑈𝑈𝑇) → 𝑈 = 𝑇) ↔ ¬ ((𝑅𝑈𝑈𝑇) ∧ ¬ 𝑈 = 𝑇))
12 anass 680 . . . . . 6 (((𝑅𝑈𝑈𝑇) ∧ ¬ 𝑈 = 𝑇) ↔ (𝑅𝑈 ∧ (𝑈𝑇 ∧ ¬ 𝑈 = 𝑇)))
13 dfpss2 3676 . . . . . . 7 (𝑈𝑇 ↔ (𝑈𝑇 ∧ ¬ 𝑈 = 𝑇))
1413anbi2i 729 . . . . . 6 ((𝑅𝑈𝑈𝑇) ↔ (𝑅𝑈 ∧ (𝑈𝑇 ∧ ¬ 𝑈 = 𝑇)))
1512, 14bitr4i 267 . . . . 5 (((𝑅𝑈𝑈𝑇) ∧ ¬ 𝑈 = 𝑇) ↔ (𝑅𝑈𝑈𝑇))
1615notbii 310 . . . 4 (¬ ((𝑅𝑈𝑈𝑇) ∧ ¬ 𝑈 = 𝑇) ↔ ¬ (𝑅𝑈𝑈𝑇))
1711, 16bitr2i 265 . . 3 (¬ (𝑅𝑈𝑈𝑇) ↔ ((𝑅𝑈𝑈𝑇) → 𝑈 = 𝑇))
1810, 17sylib 208 . 2 (𝜑 → ((𝑅𝑈𝑈𝑇) → 𝑈 = 𝑇))
191, 2, 18mp2and 714 1 (𝜑𝑈 = 𝑇)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384   = wceq 1480  wcel 1987  wss 3560  wpss 3561   class class class wbr 4623  cfv 5857  LSubSpclss 18872  L clcv 33824
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-iota 5820  df-fun 5859  df-fv 5865  df-lcv 33825
This theorem is referenced by:  lcvat  33836  lsatexch  33849
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