Theorem lcvnbtwn2 36323

Description: The covers relation implies no in-betweenness. (cvnbtwn2 30070 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 36321	. . 3 ⊢ (𝜑 → ¬ (𝑅 ⊊ 𝑈 ∧ 𝑈 ⊊ 𝑇))
11		iman 405	. . . 4 ⊢ (((𝑅 ⊊ 𝑈 ∧ 𝑈 ⊆ 𝑇) → 𝑈 = 𝑇) ↔ ¬ ((𝑅 ⊊ 𝑈 ∧ 𝑈 ⊆ 𝑇) ∧ ¬ 𝑈 = 𝑇))
12		anass 472	. . . . . 6 ⊢ (((𝑅 ⊊ 𝑈 ∧ 𝑈 ⊆ 𝑇) ∧ ¬ 𝑈 = 𝑇) ↔ (𝑅 ⊊ 𝑈 ∧ (𝑈 ⊆ 𝑇 ∧ ¬ 𝑈 = 𝑇)))
13		dfpss2 4013	. . . . . . 7 ⊢ (𝑈 ⊊ 𝑇 ↔ (𝑈 ⊆ 𝑇 ∧ ¬ 𝑈 = 𝑇))
14	13	anbi2i 625	. . . . . 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 698	1 ⊢ (𝜑 → 𝑈 = 𝑇)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111   ⊆ wss 3881   ⊊ wpss 3882   class class class wbr 5030  ‘cfv 6324  LSubSpclss 19696   ⋖_L clcv 36314

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-iota 6283  df-fun 6326  df-fv 6332  df-lcv 36315

This theorem is referenced by:  lcvat  36326  lsatexch  36339