Theorem lcvnbtwn2 38499

Description: The covers relation implies no in-betweenness. (cvnbtwn2 32110 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 38497	. . 3 ⊢ (𝜑 → ¬ (𝑅 ⊊ 𝑈 ∧ 𝑈 ⊊ 𝑇))
11		iman 401	. . . 4 ⊢ (((𝑅 ⊊ 𝑈 ∧ 𝑈 ⊆ 𝑇) → 𝑈 = 𝑇) ↔ ¬ ((𝑅 ⊊ 𝑈 ∧ 𝑈 ⊆ 𝑇) ∧ ¬ 𝑈 = 𝑇))
12		anass 468	. . . . . 6 ⊢ (((𝑅 ⊊ 𝑈 ∧ 𝑈 ⊆ 𝑇) ∧ ¬ 𝑈 = 𝑇) ↔ (𝑅 ⊊ 𝑈 ∧ (𝑈 ⊆ 𝑇 ∧ ¬ 𝑈 = 𝑇)))
13		dfpss2 4083	. . . . . . 7 ⊢ (𝑈 ⊊ 𝑇 ↔ (𝑈 ⊆ 𝑇 ∧ ¬ 𝑈 = 𝑇))
14	13	anbi2i 622	. . . . . 6 ⊢ ((𝑅 ⊊ 𝑈 ∧ 𝑈 ⊊ 𝑇) ↔ (𝑅 ⊊ 𝑈 ∧ (𝑈 ⊆ 𝑇 ∧ ¬ 𝑈 = 𝑇)))
15	12, 14	bitr4i 278	. . . . 5 ⊢ (((𝑅 ⊊ 𝑈 ∧ 𝑈 ⊆ 𝑇) ∧ ¬ 𝑈 = 𝑇) ↔ (𝑅 ⊊ 𝑈 ∧ 𝑈 ⊊ 𝑇))
16	15	notbii 320	. . . 4 ⊢ (¬ ((𝑅 ⊊ 𝑈 ∧ 𝑈 ⊆ 𝑇) ∧ ¬ 𝑈 = 𝑇) ↔ ¬ (𝑅 ⊊ 𝑈 ∧ 𝑈 ⊊ 𝑇))
17	11, 16	bitr2i 276	. . 3 ⊢ (¬ (𝑅 ⊊ 𝑈 ∧ 𝑈 ⊊ 𝑇) ↔ ((𝑅 ⊊ 𝑈 ∧ 𝑈 ⊆ 𝑇) → 𝑈 = 𝑇))
18	10, 17	sylib 217	. 2 ⊢ (𝜑 → ((𝑅 ⊊ 𝑈 ∧ 𝑈 ⊆ 𝑇) → 𝑈 = 𝑇))
19	1, 2, 18	mp2and 698	1 ⊢ (𝜑 → 𝑈 = 𝑇)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1534   ∈ wcel 2099   ⊆ wss 3947   ⊊ wpss 3948   class class class wbr 5148  ‘cfv 6548  LSubSpclss 20815   ⋖_L clcv 38490

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-iota 6500  df-fun 6550  df-fv 6556  df-lcv 38491

This theorem is referenced by:  lcvat  38502  lsatexch  38515