Theorem lcvnbtwn2 39861

Description: The covers relation implies no in-betweenness. (cvnbtwn2 32712 analog.) (Contributed by NM, 7-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)
lcvnbtwn2.p	⊢ (𝜑 → 𝑅 ⊊ 𝑈)
lcvnbtwn2.q	⊢ (𝜑 → 𝑈 ⊆ 𝑇)

Assertion

Ref	Expression
lcvnbtwn2	⊢ (𝜑 → 𝑈 = 𝑇)

Proof of Theorem lcvnbtwn2

Step	Hyp	Ref	Expression
1		lcvnbtwn2.p	. 2 ⊢ (𝜑 → 𝑅 ⊊ 𝑈)
2		lcvnbtwn2.q	. 2 ⊢ (𝜑 → 𝑈 ⊆ 𝑇)
3		lcvnbtwn.s	. . . 4 ⊢ 𝑆 = (LSubSp‘𝑊)
4		lcvnbtwn.c	. . . 4 ⊢ 𝐶 = ( ⋖L ‘𝑊)
5		lcvnbtwn.w	. . . 4 ⊢ (𝜑 → 𝑊 ∈ 𝑋)
6		lcvnbtwn.r	. . . 4 ⊢ (𝜑 → 𝑅 ∈ 𝑆)
7		lcvnbtwn.t	. . . 4 ⊢ (𝜑 → 𝑇 ∈ 𝑆)
8		lcvnbtwn.u	. . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝑆)
9		lcvnbtwn.d	. . . 4 ⊢ (𝜑 → 𝑅𝐶𝑇)
10	3, 4, 5, 6, 7, 8, 9	lcvnbtwn 39859	. . 3 ⊢ (𝜑 → ¬ (𝑅 ⊊ 𝑈 ∧ 𝑈 ⊊ 𝑇))
11		iman 407	. . . 4 ⊢ (((𝑅 ⊊ 𝑈 ∧ 𝑈 ⊆ 𝑇) → 𝑈 = 𝑇) ↔ ¬ ((𝑅 ⊊ 𝑈 ∧ 𝑈 ⊆ 𝑇) ∧ ¬ 𝑈 = 𝑇))
12		anass 474	. . . . . 6 ⊢ (((𝑅 ⊊ 𝑈 ∧ 𝑈 ⊆ 𝑇) ∧ ¬ 𝑈 = 𝑇) ↔ (𝑅 ⊊ 𝑈 ∧ (𝑈 ⊆ 𝑇 ∧ ¬ 𝑈 = 𝑇)))
13		dfpss2 4043	. . . . . . 7 ⊢ (𝑈 ⊊ 𝑇 ↔ (𝑈 ⊆ 𝑇 ∧ ¬ 𝑈 = 𝑇))
14	13	anbi2i 635	. . . . . 6 ⊢ ((𝑅 ⊊ 𝑈 ∧ 𝑈 ⊊ 𝑇) ↔ (𝑅 ⊊ 𝑈 ∧ (𝑈 ⊆ 𝑇 ∧ ¬ 𝑈 = 𝑇)))
15	12, 14	bitr4i 281	. . . . 5 ⊢ (((𝑅 ⊊ 𝑈 ∧ 𝑈 ⊆ 𝑇) ∧ ¬ 𝑈 = 𝑇) ↔ (𝑅 ⊊ 𝑈 ∧ 𝑈 ⊊ 𝑇))
16	15	notbii 323	. . . 4 ⊢ (¬ ((𝑅 ⊊ 𝑈 ∧ 𝑈 ⊆ 𝑇) ∧ ¬ 𝑈 = 𝑇) ↔ ¬ (𝑅 ⊊ 𝑈 ∧ 𝑈 ⊊ 𝑇))
17	11, 16	bitr2i 279	. . 3 ⊢ (¬ (𝑅 ⊊ 𝑈 ∧ 𝑈 ⊊ 𝑇) ↔ ((𝑅 ⊊ 𝑈 ∧ 𝑈 ⊆ 𝑇) → 𝑈 = 𝑇))
18	10, 17	sylib 221	. 2 ⊢ (𝜑 → ((𝑅 ⊊ 𝑈 ∧ 𝑈 ⊆ 𝑇) → 𝑈 = 𝑇))
19	1, 2, 18	mp2and 712	1 ⊢ (𝜑 → 𝑈 = 𝑇)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2146   ⊆ wss 3906   ⊊ wpss 3907   class class class wbr 5111  ‘cfv 6540  LSubSpclss 21104   ⋖_L clcv 39852

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-iota 6496  df-fun 6542  df-fv 6548  df-lcv 39853

This theorem is used by:  lcvat  39864  lsatexch  39877