Theorem lcvntr 39008

Description: The covers relation is not transitive. (cvntr 32321 analog.) (Contributed by NM, 10-Jan-2015.)

Hypotheses

Ref	Expression
lcvnbtwn.s	⊢ 𝑆 = (LSubSp‘𝑊)
lcvnbtwn.c	⊢ 𝐶 = ( ⋖L ‘𝑊)
lcvnbtwn.w	⊢ (𝜑 → 𝑊 ∈ 𝑋)
lcvnbtwn.r	⊢ (𝜑 → 𝑅 ∈ 𝑆)
lcvnbtwn.t	⊢ (𝜑 → 𝑇 ∈ 𝑆)
lcvnbtwn.u	⊢ (𝜑 → 𝑈 ∈ 𝑆)
lcvnbtwn.d	⊢ (𝜑 → 𝑅𝐶𝑇)
lcvntr.p	⊢ (𝜑 → 𝑇𝐶𝑈)

Assertion

Ref	Expression
lcvntr	⊢ (𝜑 → ¬ 𝑅𝐶𝑈)

Proof of Theorem lcvntr

Step	Hyp	Ref	Expression
1		lcvnbtwn.s	. . . 4 ⊢ 𝑆 = (LSubSp‘𝑊)
2		lcvnbtwn.c	. . . 4 ⊢ 𝐶 = ( ⋖L ‘𝑊)
3		lcvnbtwn.w	. . . 4 ⊢ (𝜑 → 𝑊 ∈ 𝑋)
4		lcvnbtwn.r	. . . 4 ⊢ (𝜑 → 𝑅 ∈ 𝑆)
5		lcvnbtwn.t	. . . 4 ⊢ (𝜑 → 𝑇 ∈ 𝑆)
6		lcvnbtwn.d	. . . 4 ⊢ (𝜑 → 𝑅𝐶𝑇)
7	1, 2, 3, 4, 5, 6	lcvpss 39006	. . 3 ⊢ (𝜑 → 𝑅 ⊊ 𝑇)
8		lcvnbtwn.u	. . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝑆)
9		lcvntr.p	. . . 4 ⊢ (𝜑 → 𝑇𝐶𝑈)
10	1, 2, 3, 5, 8, 9	lcvpss 39006	. . 3 ⊢ (𝜑 → 𝑇 ⊊ 𝑈)
11	7, 10	jca 511	. 2 ⊢ (𝜑 → (𝑅 ⊊ 𝑇 ∧ 𝑇 ⊊ 𝑈))
12	3	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑅𝐶𝑈) → 𝑊 ∈ 𝑋)
13	4	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑅𝐶𝑈) → 𝑅 ∈ 𝑆)
14	8	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑅𝐶𝑈) → 𝑈 ∈ 𝑆)
15	5	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑅𝐶𝑈) → 𝑇 ∈ 𝑆)
16		simpr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑅𝐶𝑈) → 𝑅𝐶𝑈)
17	1, 2, 12, 13, 14, 15, 16	lcvnbtwn 39007	. . 3 ⊢ ((𝜑 ∧ 𝑅𝐶𝑈) → ¬ (𝑅 ⊊ 𝑇 ∧ 𝑇 ⊊ 𝑈))
18	17	ex 412	. 2 ⊢ (𝜑 → (𝑅𝐶𝑈 → ¬ (𝑅 ⊊ 𝑇 ∧ 𝑇 ⊊ 𝑈)))
19	11, 18	mt2d 136	1 ⊢ (𝜑 → ¬ 𝑅𝐶𝑈)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2106   ⊊ wpss 3964   class class class wbr 5148  ‘cfv 6563  LSubSpclss 20947   ⋖_L clcv 39000

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-lcv 39001

This theorem is referenced by:  lsatcv0eq  39029