lcvntr - Mathbox for Norm Megill

	Mathbox for Norm Megill		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > lcvntr		Structured version Visualization version GIF version

Theorem lcvntr 39860

Description: The covers relation is not transitive. (cvntr 32717 analog.) (Contributed by NM, 10-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)
lcvntr.p	⊢ (𝜑 → 𝑇𝐶𝑈)

**Assertion**
Ref	Expression
lcvntr	⊢ (𝜑 → ¬ 𝑅𝐶𝑈)

**Proof of Theorem lcvntr**
Step	Hyp	Ref	Expression
1		lcvnbtwn.s	. . . 4 ⊢ 𝑆 = (LSubSp‘𝑊)
2		lcvnbtwn.c	. . . 4 ⊢ 𝐶 = ( ⋖_L ‘𝑊)
3		lcvnbtwn.w	. . . 4 ⊢ (𝜑 → 𝑊 ∈ 𝑋)
4		lcvnbtwn.r	. . . 4 ⊢ (𝜑 → 𝑅 ∈ 𝑆)
5		lcvnbtwn.t	. . . 4 ⊢ (𝜑 → 𝑇 ∈ 𝑆)
6		lcvnbtwn.d	. . . 4 ⊢ (𝜑 → 𝑅𝐶𝑇)
7	1, 2, 3, 4, 5, 6	lcvpss 39858	. . 3 ⊢ (𝜑 → 𝑅 ⊊ 𝑇)
8		lcvnbtwn.u	. . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝑆)
9		lcvntr.p	. . . 4 ⊢ (𝜑 → 𝑇𝐶𝑈)
10	1, 2, 3, 5, 8, 9	lcvpss 39858	. . 3 ⊢ (𝜑 → 𝑇 ⊊ 𝑈)
11	7, 10	jca 521	. 2 ⊢ (𝜑 → (𝑅 ⊊ 𝑇 ∧ 𝑇 ⊊ 𝑈))
12	3	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑅𝐶𝑈) → 𝑊 ∈ 𝑋)
13	4	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑅𝐶𝑈) → 𝑅 ∈ 𝑆)
14	8	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑅𝐶𝑈) → 𝑈 ∈ 𝑆)
15	5	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑅𝐶𝑈) → 𝑇 ∈ 𝑆)
16		simpr 490	. . . 4 ⊢ ((𝜑 ∧ 𝑅𝐶𝑈) → 𝑅𝐶𝑈)
17	1, 2, 12, 13, 14, 15, 16	lcvnbtwn 39859	. . 3 ⊢ ((𝜑 ∧ 𝑅𝐶𝑈) → ¬ (𝑅 ⊊ 𝑇 ∧ 𝑇 ⊊ 𝑈))
18	17	ex 418	. 2 ⊢ (𝜑 → (𝑅𝐶𝑈 → ¬ (𝑅 ⊊ 𝑇 ∧ 𝑇 ⊊ 𝑈)))
19	11, 18	mt2d 137	1 ⊢ (𝜑 → ¬ 𝑅𝐶𝑈)

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2146 ⊊ 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: lsatcv0eq 39881

W3C validator