Theorem lcvnbtwn 40082

Description: The covers relation implies no in-betweenness. (cvnbtwn 32888 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 40078	. . . 4 ⊢ (𝜑 → (𝑅𝐶𝑇 ↔ (𝑅 ⊊ 𝑇 ∧ ¬ ∃𝑢 ∈ 𝑆 (𝑅 ⊊ 𝑢 ∧ 𝑢 ⊊ 𝑇))))
8	1, 7	mpbid 235	. . 3 ⊢ (𝜑 → (𝑅 ⊊ 𝑇 ∧ ¬ ∃𝑢 ∈ 𝑆 (𝑅 ⊊ 𝑢 ∧ 𝑢 ⊊ 𝑇)))
9	8	simprd 501	. 2 ⊢ (𝜑 → ¬ ∃𝑢 ∈ 𝑆 (𝑅 ⊊ 𝑢 ∧ 𝑢 ⊊ 𝑇))
10		lcvnbtwn.u	. . 3 ⊢ (𝜑 → 𝑈 ∈ 𝑆)
11		psseq2 4039	. . . . 5 ⊢ (𝑢 = 𝑈 → (𝑅 ⊊ 𝑢 ↔ 𝑅 ⊊ 𝑈))
12		psseq1 4038	. . . . 5 ⊢ (𝑢 = 𝑈 → (𝑢 ⊊ 𝑇 ↔ 𝑈 ⊊ 𝑇))
13	11, 12	anbi12d 644	. . . 4 ⊢ (𝑢 = 𝑈 → ((𝑅 ⊊ 𝑢 ∧ 𝑢 ⊊ 𝑇) ↔ (𝑅 ⊊ 𝑈 ∧ 𝑈 ⊊ 𝑇)))
14	13	rspcev 3577	. . 3 ⊢ ((𝑈 ∈ 𝑆 ∧ (𝑅 ⊊ 𝑈 ∧ 𝑈 ⊊ 𝑇)) → ∃𝑢 ∈ 𝑆 (𝑅 ⊊ 𝑢 ∧ 𝑢 ⊊ 𝑇))
15	10, 14	sylan 592	. 2 ⊢ ((𝜑 ∧ (𝑅 ⊊ 𝑈 ∧ 𝑈 ⊊ 𝑇)) → ∃𝑢 ∈ 𝑆 (𝑅 ⊊ 𝑢 ∧ 𝑢 ⊊ 𝑇))
16	9, 15	mtand 828	1 ⊢ (𝜑 → ¬ (𝑅 ⊊ 𝑈 ∧ 𝑈 ⊊ 𝑇))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∃wrex 3087   ⊊ wpss 3900   class class class wbr 5103  ‘cfv 6538  LSubSpclss 21206   ⋖_L clcv 40075

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751

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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-iota 6494  df-fun 6540  df-fv 6546  df-lcv 40076

This theorem is used by:  lcvntr  40083  lcvnbtwn2  40084  lcvnbtwn3  40085