Theorem cvrnbtwn4 40053

Description: The covers relation implies no in-betweenness. Part of proof of Lemma 7.5.1 of [MaedaMaeda] p. 31. (cvnbtwn4 32641 analog.) (Contributed by NM, 18-Oct-2011.)

Hypotheses

Ref	Expression
cvrle.b	⊢ 𝐵 = (Base‘𝐾)
cvrle.l	⊢ ≤ = (le‘𝐾)
cvrle.c	⊢ 𝐶 = ( ⋖ ‘𝐾)

Assertion

Ref	Expression
cvrnbtwn4	⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → ((𝑋 ≤ 𝑍 ∧ 𝑍 ≤ 𝑌) ↔ (𝑋 = 𝑍 ∨ 𝑍 = 𝑌)))

Proof of Theorem cvrnbtwn4

Step	Hyp	Ref	Expression
1		cvrle.b	. . . 4 ⊢ 𝐵 = (Base‘𝐾)
2		eqid 2763	. . . 4 ⊢ (lt‘𝐾) = (lt‘𝐾)
3		cvrle.c	. . . 4 ⊢ 𝐶 = ( ⋖ ‘𝐾)
4	1, 2, 3	cvrnbtwn 40045	. . 3 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → ¬ (𝑋(lt‘𝐾)𝑍 ∧ 𝑍(lt‘𝐾)𝑌))
5		iman 406	. . . . 5 ⊢ (((𝑋 ≤ 𝑍 ∧ 𝑍 ≤ 𝑌) → (𝑋 = 𝑍 ∨ 𝑍 = 𝑌)) ↔ ¬ ((𝑋 ≤ 𝑍 ∧ 𝑍 ≤ 𝑌) ∧ ¬ (𝑋 = 𝑍 ∨ 𝑍 = 𝑌)))
6		neanior 3051	. . . . . . . . 9 ⊢ ((𝑋 ≠ 𝑍 ∧ 𝑍 ≠ 𝑌) ↔ ¬ (𝑋 = 𝑍 ∨ 𝑍 = 𝑌))
7	6	anbi2i 634	. . . . . . . 8 ⊢ (((𝑋 ≤ 𝑍 ∧ 𝑍 ≤ 𝑌) ∧ (𝑋 ≠ 𝑍 ∧ 𝑍 ≠ 𝑌)) ↔ ((𝑋 ≤ 𝑍 ∧ 𝑍 ≤ 𝑌) ∧ ¬ (𝑋 = 𝑍 ∨ 𝑍 = 𝑌)))
8		an4 668	. . . . . . . 8 ⊢ (((𝑋 ≤ 𝑍 ∧ 𝑍 ≤ 𝑌) ∧ (𝑋 ≠ 𝑍 ∧ 𝑍 ≠ 𝑌)) ↔ ((𝑋 ≤ 𝑍 ∧ 𝑋 ≠ 𝑍) ∧ (𝑍 ≤ 𝑌 ∧ 𝑍 ≠ 𝑌)))
9	7, 8	bitr3i 280	. . . . . . 7 ⊢ (((𝑋 ≤ 𝑍 ∧ 𝑍 ≤ 𝑌) ∧ ¬ (𝑋 = 𝑍 ∨ 𝑍 = 𝑌)) ↔ ((𝑋 ≤ 𝑍 ∧ 𝑋 ≠ 𝑍) ∧ (𝑍 ≤ 𝑌 ∧ 𝑍 ≠ 𝑌)))
10		cvrle.l	. . . . . . . . . 10 ⊢ ≤ = (le‘𝐾)
11	10, 2	pltval 18381	. . . . . . . . 9 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (𝑋(lt‘𝐾)𝑍 ↔ (𝑋 ≤ 𝑍 ∧ 𝑋 ≠ 𝑍)))
12	11	3adant3r2 1202	. . . . . . . 8 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋(lt‘𝐾)𝑍 ↔ (𝑋 ≤ 𝑍 ∧ 𝑋 ≠ 𝑍)))
13	10, 2	pltval 18381	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Poset ∧ 𝑍 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑍(lt‘𝐾)𝑌 ↔ (𝑍 ≤ 𝑌 ∧ 𝑍 ≠ 𝑌)))
14	13	3com23 1144	. . . . . . . . 9 ⊢ ((𝐾 ∈ Poset ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (𝑍(lt‘𝐾)𝑌 ↔ (𝑍 ≤ 𝑌 ∧ 𝑍 ≠ 𝑌)))
15	14	3adant3r1 1201	. . . . . . . 8 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑍(lt‘𝐾)𝑌 ↔ (𝑍 ≤ 𝑌 ∧ 𝑍 ≠ 𝑌)))
16	12, 15	anbi12d 643	. . . . . . 7 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋(lt‘𝐾)𝑍 ∧ 𝑍(lt‘𝐾)𝑌) ↔ ((𝑋 ≤ 𝑍 ∧ 𝑋 ≠ 𝑍) ∧ (𝑍 ≤ 𝑌 ∧ 𝑍 ≠ 𝑌))))
17	9, 16	bitr4id 293	. . . . . 6 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (((𝑋 ≤ 𝑍 ∧ 𝑍 ≤ 𝑌) ∧ ¬ (𝑋 = 𝑍 ∨ 𝑍 = 𝑌)) ↔ (𝑋(lt‘𝐾)𝑍 ∧ 𝑍(lt‘𝐾)𝑌)))
18	17	notbid 321	. . . . 5 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (¬ ((𝑋 ≤ 𝑍 ∧ 𝑍 ≤ 𝑌) ∧ ¬ (𝑋 = 𝑍 ∨ 𝑍 = 𝑌)) ↔ ¬ (𝑋(lt‘𝐾)𝑍 ∧ 𝑍(lt‘𝐾)𝑌)))
19	5, 18	bitr2id 287	. . . 4 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (¬ (𝑋(lt‘𝐾)𝑍 ∧ 𝑍(lt‘𝐾)𝑌) ↔ ((𝑋 ≤ 𝑍 ∧ 𝑍 ≤ 𝑌) → (𝑋 = 𝑍 ∨ 𝑍 = 𝑌))))
20	19	3adant3 1150	. . 3 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → (¬ (𝑋(lt‘𝐾)𝑍 ∧ 𝑍(lt‘𝐾)𝑌) ↔ ((𝑋 ≤ 𝑍 ∧ 𝑍 ≤ 𝑌) → (𝑋 = 𝑍 ∨ 𝑍 = 𝑌))))
21	4, 20	mpbid 235	. 2 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → ((𝑋 ≤ 𝑍 ∧ 𝑍 ≤ 𝑌) → (𝑋 = 𝑍 ∨ 𝑍 = 𝑌)))
22	1, 10	posref 18369	. . . . . . 7 ⊢ ((𝐾 ∈ Poset ∧ 𝑍 ∈ 𝐵) → 𝑍 ≤ 𝑍)
23	22	3ad2antr3 1209	. . . . . 6 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝑍 ≤ 𝑍)
24	23	3adant3 1150	. . . . 5 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → 𝑍 ≤ 𝑍)
25		breq1 5112	. . . . 5 ⊢ (𝑋 = 𝑍 → (𝑋 ≤ 𝑍 ↔ 𝑍 ≤ 𝑍))
26	24, 25	syl5ibrcom 250	. . . 4 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → (𝑋 = 𝑍 → 𝑋 ≤ 𝑍))
27	1, 10, 3	cvrle 40052	. . . . . . . 8 ⊢ (((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → 𝑋 ≤ 𝑌)
28	27	ex 417	. . . . . . 7 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐶𝑌 → 𝑋 ≤ 𝑌))
29	28	3adant3r3 1203	. . . . . 6 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋𝐶𝑌 → 𝑋 ≤ 𝑌))
30	29	3impia 1135	. . . . 5 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → 𝑋 ≤ 𝑌)
31		breq2 5113	. . . . 5 ⊢ (𝑍 = 𝑌 → (𝑋 ≤ 𝑍 ↔ 𝑋 ≤ 𝑌))
32	30, 31	syl5ibrcom 250	. . . 4 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → (𝑍 = 𝑌 → 𝑋 ≤ 𝑍))
33	26, 32	jaod 872	. . 3 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → ((𝑋 = 𝑍 ∨ 𝑍 = 𝑌) → 𝑋 ≤ 𝑍))
34		breq1 5112	. . . . 5 ⊢ (𝑋 = 𝑍 → (𝑋 ≤ 𝑌 ↔ 𝑍 ≤ 𝑌))
35	30, 34	syl5ibcom 248	. . . 4 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → (𝑋 = 𝑍 → 𝑍 ≤ 𝑌))
36		breq2 5113	. . . . 5 ⊢ (𝑍 = 𝑌 → (𝑍 ≤ 𝑍 ↔ 𝑍 ≤ 𝑌))
37	24, 36	syl5ibcom 248	. . . 4 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → (𝑍 = 𝑌 → 𝑍 ≤ 𝑌))
38	35, 37	jaod 872	. . 3 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → ((𝑋 = 𝑍 ∨ 𝑍 = 𝑌) → 𝑍 ≤ 𝑌))
39	33, 38	jcad 521	. 2 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → ((𝑋 = 𝑍 ∨ 𝑍 = 𝑌) → (𝑋 ≤ 𝑍 ∧ 𝑍 ≤ 𝑌)))
40	21, 39	impbid 215	1 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → ((𝑋 ≤ 𝑍 ∧ 𝑍 ≤ 𝑌) ↔ (𝑋 = 𝑍 ∨ 𝑍 = 𝑌)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143   ≠ wne 2958   class class class wbr 5109  ‘cfv 6536  Basecbs 17264  lecple 17312  Posetcpo 18358  ltcplt 18359   ⋖ ccvr 40036

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3745  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-iota 6492  df-fun 6538  df-fv 6544  df-proset 18345  df-poset 18364  df-plt 18379  df-covers 40040

This theorem is referenced by:  cvrcmp  40057  leatb  40066  2llnmat  40298  2lnat  40558