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Theorem lcvnbtwn 34630
Description: The covers relation implies no in-betweenness. (cvnbtwn 29273 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)
Assertion
Ref Expression
lcvnbtwn (𝜑 → ¬ (𝑅𝑈𝑈𝑇))

Proof of Theorem lcvnbtwn
Dummy variable 𝑢 is distinct from all other variables.
StepHypRef Expression
1 lcvnbtwn.d . . . 4 (𝜑𝑅𝐶𝑇)
2 lcvnbtwn.s . . . . 5 𝑆 = (LSubSp‘𝑊)
3 lcvnbtwn.c . . . . 5 𝐶 = ( ⋖L𝑊)
4 lcvnbtwn.w . . . . 5 (𝜑𝑊𝑋)
5 lcvnbtwn.r . . . . 5 (𝜑𝑅𝑆)
6 lcvnbtwn.t . . . . 5 (𝜑𝑇𝑆)
72, 3, 4, 5, 6lcvbr 34626 . . . 4 (𝜑 → (𝑅𝐶𝑇 ↔ (𝑅𝑇 ∧ ¬ ∃𝑢𝑆 (𝑅𝑢𝑢𝑇))))
81, 7mpbid 222 . . 3 (𝜑 → (𝑅𝑇 ∧ ¬ ∃𝑢𝑆 (𝑅𝑢𝑢𝑇)))
98simprd 478 . 2 (𝜑 → ¬ ∃𝑢𝑆 (𝑅𝑢𝑢𝑇))
10 lcvnbtwn.u . . 3 (𝜑𝑈𝑆)
11 psseq2 3728 . . . . 5 (𝑢 = 𝑈 → (𝑅𝑢𝑅𝑈))
12 psseq1 3727 . . . . 5 (𝑢 = 𝑈 → (𝑢𝑇𝑈𝑇))
1311, 12anbi12d 747 . . . 4 (𝑢 = 𝑈 → ((𝑅𝑢𝑢𝑇) ↔ (𝑅𝑈𝑈𝑇)))
1413rspcev 3340 . . 3 ((𝑈𝑆 ∧ (𝑅𝑈𝑈𝑇)) → ∃𝑢𝑆 (𝑅𝑢𝑢𝑇))
1510, 14sylan 487 . 2 ((𝜑 ∧ (𝑅𝑈𝑈𝑇)) → ∃𝑢𝑆 (𝑅𝑢𝑢𝑇))
169, 15mtand 692 1 (𝜑 → ¬ (𝑅𝑈𝑈𝑇))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383   = wceq 1523  wcel 2030  wrex 2942  wpss 3608   class class class wbr 4685  cfv 5926  LSubSpclss 18980  L clcv 34623
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fv 5934  df-lcv 34624
This theorem is referenced by:  lcvntr  34631  lcvnbtwn2  34632  lcvnbtwn3  34633
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