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Theorem lcvnbtwn 36594
 Description: The covers relation implies no in-betweenness. (cvnbtwn 30161 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 36590 . . . 4 (𝜑 → (𝑅𝐶𝑇 ↔ (𝑅𝑇 ∧ ¬ ∃𝑢𝑆 (𝑅𝑢𝑢𝑇))))
81, 7mpbid 235 . . 3 (𝜑 → (𝑅𝑇 ∧ ¬ ∃𝑢𝑆 (𝑅𝑢𝑢𝑇)))
98simprd 500 . 2 (𝜑 → ¬ ∃𝑢𝑆 (𝑅𝑢𝑢𝑇))
10 lcvnbtwn.u . . 3 (𝜑𝑈𝑆)
11 psseq2 3995 . . . . 5 (𝑢 = 𝑈 → (𝑅𝑢𝑅𝑈))
12 psseq1 3994 . . . . 5 (𝑢 = 𝑈 → (𝑢𝑇𝑈𝑇))
1311, 12anbi12d 634 . . . 4 (𝑢 = 𝑈 → ((𝑅𝑢𝑢𝑇) ↔ (𝑅𝑈𝑈𝑇)))
1413rspcev 3542 . . 3 ((𝑈𝑆 ∧ (𝑅𝑈𝑈𝑇)) → ∃𝑢𝑆 (𝑅𝑢𝑢𝑇))
1510, 14sylan 584 . 2 ((𝜑 ∧ (𝑅𝑈𝑈𝑇)) → ∃𝑢𝑆 (𝑅𝑢𝑢𝑇))
169, 15mtand 816 1 (𝜑 → ¬ (𝑅𝑈𝑈𝑇))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   = wceq 1539   ∈ wcel 2112  ∃wrex 3072   ⊊ wpss 3860   class class class wbr 5033  ‘cfv 6336  LSubSpclss 19764   ⋖L clcv 36587 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 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5170  ax-nul 5177  ax-pow 5235  ax-pr 5299  ax-un 7460 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rex 3077  df-rab 3080  df-v 3412  df-sbc 3698  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-pss 3878  df-nul 4227  df-if 4422  df-pw 4497  df-sn 4524  df-pr 4526  df-op 4530  df-uni 4800  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5431  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-iota 6295  df-fun 6338  df-fv 6344  df-lcv 36588 This theorem is referenced by:  lcvntr  36595  lcvnbtwn2  36596  lcvnbtwn3  36597
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