lcvnbtwn3 - Mathbox for Norm Megill

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Theorem lcvnbtwn3 39028

Description: The covers relation implies no in-betweenness. (cvnbtwn3 32224 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 39025	. . 3 ⊢ (𝜑 → ¬ (𝑅 ⊊ 𝑈 ∧ 𝑈 ⊊ 𝑇))
11		iman 401	. . . 4 ⊢ (((𝑅 ⊆ 𝑈 ∧ 𝑈 ⊊ 𝑇) → 𝑅 = 𝑈) ↔ ¬ ((𝑅 ⊆ 𝑈 ∧ 𝑈 ⊊ 𝑇) ∧ ¬ 𝑅 = 𝑈))
12		eqcom 2737	. . . . 5 ⊢ (𝑈 = 𝑅 ↔ 𝑅 = 𝑈)
13	12	imbi2i 336	. . . 4 ⊢ (((𝑅 ⊆ 𝑈 ∧ 𝑈 ⊊ 𝑇) → 𝑈 = 𝑅) ↔ ((𝑅 ⊆ 𝑈 ∧ 𝑈 ⊊ 𝑇) → 𝑅 = 𝑈))
14		dfpss2 4054	. . . . . . 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 1540 ∈ wcel 2109 ⊆ wss 3917 ⊊ wpss 3918 class class class wbr 5110 ‘cfv 6514 LSubSpclss 20844 ⋖_L clcv 39018

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-ral 3046 df-rex 3055 df-rab 3409 df-v 3452 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-br 5111 df-opab 5173 df-mpt 5192 df-id 5536 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-iota 6467 df-fun 6516 df-fv 6522 df-lcv 39019

This theorem is referenced by: lsatcveq0 39032 lsatcvatlem 39049

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