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Theorem lcvnbtwn 37764
Description: The covers relation implies no in-betweenness. (cvnbtwn 31466 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 37760 . . . 4 (𝜑 → (𝑅𝐶𝑇 ↔ (𝑅𝑇 ∧ ¬ ∃𝑢𝑆 (𝑅𝑢𝑢𝑇))))
81, 7mpbid 231 . . 3 (𝜑 → (𝑅𝑇 ∧ ¬ ∃𝑢𝑆 (𝑅𝑢𝑢𝑇)))
98simprd 496 . 2 (𝜑 → ¬ ∃𝑢𝑆 (𝑅𝑢𝑢𝑇))
10 lcvnbtwn.u . . 3 (𝜑𝑈𝑆)
11 psseq2 4085 . . . . 5 (𝑢 = 𝑈 → (𝑅𝑢𝑅𝑈))
12 psseq1 4084 . . . . 5 (𝑢 = 𝑈 → (𝑢𝑇𝑈𝑇))
1311, 12anbi12d 631 . . . 4 (𝑢 = 𝑈 → ((𝑅𝑢𝑢𝑇) ↔ (𝑅𝑈𝑈𝑇)))
1413rspcev 3610 . . 3 ((𝑈𝑆 ∧ (𝑅𝑈𝑈𝑇)) → ∃𝑢𝑆 (𝑅𝑢𝑢𝑇))
1510, 14sylan 580 . 2 ((𝜑 ∧ (𝑅𝑈𝑈𝑇)) → ∃𝑢𝑆 (𝑅𝑢𝑢𝑇))
169, 15mtand 814 1 (𝜑 → ¬ (𝑅𝑈𝑈𝑇))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396   = wceq 1541  wcel 2106  wrex 3070  wpss 3946   class class class wbr 5142  cfv 6533  LSubSpclss 20493  L clcv 37757
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5293  ax-nul 5300  ax-pow 5357  ax-pr 5421  ax-un 7709
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-pss 3964  df-nul 4320  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5568  df-xp 5676  df-rel 5677  df-cnv 5678  df-co 5679  df-dm 5680  df-iota 6485  df-fun 6535  df-fv 6541  df-lcv 37758
This theorem is referenced by:  lcvntr  37765  lcvnbtwn2  37766  lcvnbtwn3  37767
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