Theorem lcvntr 33331

Description: The covers relation is not transitive. (cvntr 28535 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 33329	. . 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 33329	. . 3 ⊢ (𝜑 → 𝑇 ⊊ 𝑈)
11	7, 10	jca 553	. 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 476	. . . 4 ⊢ ((𝜑 ∧ 𝑅𝐶𝑈) → 𝑅𝐶𝑈)
17	1, 2, 12, 13, 14, 15, 16	lcvnbtwn 33330	. . 3 ⊢ ((𝜑 ∧ 𝑅𝐶𝑈) → ¬ (𝑅 ⊊ 𝑇 ∧ 𝑇 ⊊ 𝑈))
18	17	ex 449	. 2 ⊢ (𝜑 → (𝑅𝐶𝑈 → ¬ (𝑅 ⊊ 𝑇 ∧ 𝑇 ⊊ 𝑈)))
19	11, 18	mt2d 130	1 ⊢ (𝜑 → ¬ 𝑅𝐶𝑈)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ⊊ wpss 3541   class class class wbr 4583  ‘cfv 5804  LSubSpclss 18753   ⋖_L clcv 33323

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-lcv 33324

This theorem is referenced by:  lsatcv0eq  33352