lcvnbtwn3 - Mathbox for Norm Megill

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Theorem lcvnbtwn3 39010

Description: The covers relation implies no in-betweenness. (cvnbtwn3 32317 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)
lcvnbtwn3.p	⊢ (𝜑 → 𝑅 ⊆ 𝑈)
lcvnbtwn3.q	⊢ (𝜑 → 𝑈 ⊊ 𝑇)

**Assertion**
Ref	Expression
lcvnbtwn3	⊢ (𝜑 → 𝑈 = 𝑅)

**Proof of Theorem lcvnbtwn3**
Step	Hyp	Ref	Expression
1		lcvnbtwn3.p	. 2 ⊢ (𝜑 → 𝑅 ⊆ 𝑈)
2		lcvnbtwn3.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 39007	. . 3 ⊢ (𝜑 → ¬ (𝑅 ⊊ 𝑈 ∧ 𝑈 ⊊ 𝑇))
11		iman 401	. . . 4 ⊢ (((𝑅 ⊆ 𝑈 ∧ 𝑈 ⊊ 𝑇) → 𝑅 = 𝑈) ↔ ¬ ((𝑅 ⊆ 𝑈 ∧ 𝑈 ⊊ 𝑇) ∧ ¬ 𝑅 = 𝑈))
12		eqcom 2742	. . . . 5 ⊢ (𝑈 = 𝑅 ↔ 𝑅 = 𝑈)
13	12	imbi2i 336	. . . 4 ⊢ (((𝑅 ⊆ 𝑈 ∧ 𝑈 ⊊ 𝑇) → 𝑈 = 𝑅) ↔ ((𝑅 ⊆ 𝑈 ∧ 𝑈 ⊊ 𝑇) → 𝑅 = 𝑈))
14		dfpss2 4098	. . . . . . 7 ⊢ (𝑅 ⊊ 𝑈 ↔ (𝑅 ⊆ 𝑈 ∧ ¬ 𝑅 = 𝑈))
15	14	anbi1i 624	. . . . . 6 ⊢ ((𝑅 ⊊ 𝑈 ∧ 𝑈 ⊊ 𝑇) ↔ ((𝑅 ⊆ 𝑈 ∧ ¬ 𝑅 = 𝑈) ∧ 𝑈 ⊊ 𝑇))
16		an32 646	. . . . . 6 ⊢ (((𝑅 ⊆ 𝑈 ∧ ¬ 𝑅 = 𝑈) ∧ 𝑈 ⊊ 𝑇) ↔ ((𝑅 ⊆ 𝑈 ∧ 𝑈 ⊊ 𝑇) ∧ ¬ 𝑅 = 𝑈))
17	15, 16	bitri 275	. . . . 5 ⊢ ((𝑅 ⊊ 𝑈 ∧ 𝑈 ⊊ 𝑇) ↔ ((𝑅 ⊆ 𝑈 ∧ 𝑈 ⊊ 𝑇) ∧ ¬ 𝑅 = 𝑈))
18	17	notbii 320	. . . 4 ⊢ (¬ (𝑅 ⊊ 𝑈 ∧ 𝑈 ⊊ 𝑇) ↔ ¬ ((𝑅 ⊆ 𝑈 ∧ 𝑈 ⊊ 𝑇) ∧ ¬ 𝑅 = 𝑈))
19	11, 13, 18	3bitr4ri 304	. . 3 ⊢ (¬ (𝑅 ⊊ 𝑈 ∧ 𝑈 ⊊ 𝑇) ↔ ((𝑅 ⊆ 𝑈 ∧ 𝑈 ⊊ 𝑇) → 𝑈 = 𝑅))
20	10, 19	sylib 218	. 2 ⊢ (𝜑 → ((𝑅 ⊆ 𝑈 ∧ 𝑈 ⊊ 𝑇) → 𝑈 = 𝑅))
21	1, 2, 20	mp2and 699	1 ⊢ (𝜑 → 𝑈 = 𝑅)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ⊆ wss 3963 ⊊ wpss 3964 class class class wbr 5148 ‘cfv 6563 LSubSpclss 20947 ⋖_L clcv 39000

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-iota 6516 df-fun 6565 df-fv 6571 df-lcv 39001

This theorem is referenced by: lsatcveq0 39014 lsatcvatlem 39031

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