Theorem lcvnbtwn 38972

Description: The covers relation implies no in-betweenness. (cvnbtwn 32201 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)

Assertion

Ref	Expression
lcvnbtwn	⊢ (𝜑 → ¬ (𝑅 ⊊ 𝑈 ∧ 𝑈 ⊊ 𝑇))

Proof of Theorem lcvnbtwn

Dummy variable 𝑢 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		lcvnbtwn.d	. . . 4 ⊢ (𝜑 → 𝑅𝐶𝑇)
2		lcvnbtwn.s	. . . . 5 ⊢ 𝑆 = (LSubSp‘𝑊)
3		lcvnbtwn.c	. . . . 5 ⊢ 𝐶 = ( ⋖L ‘𝑊)
4		lcvnbtwn.w	. . . . 5 ⊢ (𝜑 → 𝑊 ∈ 𝑋)
5		lcvnbtwn.r	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ 𝑆)
6		lcvnbtwn.t	. . . . 5 ⊢ (𝜑 → 𝑇 ∈ 𝑆)
7	2, 3, 4, 5, 6	lcvbr 38968	. . . 4 ⊢ (𝜑 → (𝑅𝐶𝑇 ↔ (𝑅 ⊊ 𝑇 ∧ ¬ ∃𝑢 ∈ 𝑆 (𝑅 ⊊ 𝑢 ∧ 𝑢 ⊊ 𝑇))))
8	1, 7	mpbid 232	. . 3 ⊢ (𝜑 → (𝑅 ⊊ 𝑇 ∧ ¬ ∃𝑢 ∈ 𝑆 (𝑅 ⊊ 𝑢 ∧ 𝑢 ⊊ 𝑇)))
9	8	simprd 495	. 2 ⊢ (𝜑 → ¬ ∃𝑢 ∈ 𝑆 (𝑅 ⊊ 𝑢 ∧ 𝑢 ⊊ 𝑇))
10		lcvnbtwn.u	. . 3 ⊢ (𝜑 → 𝑈 ∈ 𝑆)
11		psseq2 4064	. . . . 5 ⊢ (𝑢 = 𝑈 → (𝑅 ⊊ 𝑢 ↔ 𝑅 ⊊ 𝑈))
12		psseq1 4063	. . . . 5 ⊢ (𝑢 = 𝑈 → (𝑢 ⊊ 𝑇 ↔ 𝑈 ⊊ 𝑇))
13	11, 12	anbi12d 632	. . . 4 ⊢ (𝑢 = 𝑈 → ((𝑅 ⊊ 𝑢 ∧ 𝑢 ⊊ 𝑇) ↔ (𝑅 ⊊ 𝑈 ∧ 𝑈 ⊊ 𝑇)))
14	13	rspcev 3599	. . 3 ⊢ ((𝑈 ∈ 𝑆 ∧ (𝑅 ⊊ 𝑈 ∧ 𝑈 ⊊ 𝑇)) → ∃𝑢 ∈ 𝑆 (𝑅 ⊊ 𝑢 ∧ 𝑢 ⊊ 𝑇))
15	10, 14	sylan 580	. 2 ⊢ ((𝜑 ∧ (𝑅 ⊊ 𝑈 ∧ 𝑈 ⊊ 𝑇)) → ∃𝑢 ∈ 𝑆 (𝑅 ⊊ 𝑢 ∧ 𝑢 ⊊ 𝑇))
16	9, 15	mtand 815	1 ⊢ (𝜑 → ¬ (𝑅 ⊊ 𝑈 ∧ 𝑈 ⊊ 𝑇))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2107  ∃wrex 3059   ⊊ wpss 3925   class class class wbr 5117  ‘cfv 6528  LSubSpclss 20875   ⋖_L clcv 38965

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-sep 5264  ax-nul 5274  ax-pow 5333  ax-pr 5400  ax-un 7724

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rab 3414  df-v 3459  df-dif 3927  df-un 3929  df-in 3931  df-ss 3941  df-pss 3944  df-nul 4307  df-if 4499  df-pw 4575  df-sn 4600  df-pr 4602  df-op 4606  df-uni 4882  df-br 5118  df-opab 5180  df-mpt 5200  df-id 5546  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-iota 6481  df-fun 6530  df-fv 6536  df-lcv 38966

This theorem is referenced by:  lcvntr  38973  lcvnbtwn2  38974  lcvnbtwn3  38975