Theorem lcvntr 39402

Description: The covers relation is not transitive. (cvntr 32380 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 39400	. . 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 39400	. . 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 39401	. . 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 1542   ∈ wcel 2114   ⊊ wpss 3904   class class class wbr 5100  ‘cfv 6500  LSubSpclss 20894   ⋖_L clcv 39394

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-iota 6456  df-fun 6502  df-fv 6508  df-lcv 39395

This theorem is referenced by:  lsatcv0eq  39423