Theorem cvrnbtwn2 40049

Description: The covers relation implies no in-betweenness. (cvnbtwn2 32639 analog.) (Contributed by NM, 17-Nov-2011.)

Hypotheses

Ref	Expression
cvrletr.b	⊢ 𝐵 = (Base‘𝐾)
cvrletr.l	⊢ ≤ = (le‘𝐾)
cvrletr.s	⊢ < = (lt‘𝐾)
cvrletr.c	⊢ 𝐶 = ( ⋖ ‘𝐾)

Assertion

Ref	Expression
cvrnbtwn2	⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → ((𝑋 < 𝑍 ∧ 𝑍 ≤ 𝑌) ↔ 𝑍 = 𝑌))

Proof of Theorem cvrnbtwn2

Step	Hyp	Ref	Expression
1		cvrletr.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐾)
2		cvrletr.s	. . . . . 6 ⊢ < = (lt‘𝐾)
3		cvrletr.c	. . . . . 6 ⊢ 𝐶 = ( ⋖ ‘𝐾)
4	1, 2, 3	cvrnbtwn 40045	. . . . 5 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → ¬ (𝑋 < 𝑍 ∧ 𝑍 < 𝑌))
5	4	3expia 1139	. . . 4 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋𝐶𝑌 → ¬ (𝑋 < 𝑍 ∧ 𝑍 < 𝑌)))
6		iman 406	. . . . 5 ⊢ (((𝑋 < 𝑍 ∧ 𝑍 ≤ 𝑌) → 𝑍 = 𝑌) ↔ ¬ ((𝑋 < 𝑍 ∧ 𝑍 ≤ 𝑌) ∧ ¬ 𝑍 = 𝑌))
7		anass 473	. . . . . . 7 ⊢ (((𝑋 < 𝑍 ∧ 𝑍 ≤ 𝑌) ∧ ¬ 𝑍 = 𝑌) ↔ (𝑋 < 𝑍 ∧ (𝑍 ≤ 𝑌 ∧ ¬ 𝑍 = 𝑌)))
8		simpl 487	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝐾 ∈ Poset)
9		simpr3 1215	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝑍 ∈ 𝐵)
10		simpr2 1214	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝑌 ∈ 𝐵)
11		cvrletr.l	. . . . . . . . . . 11 ⊢ ≤ = (le‘𝐾)
12	11, 2	pltval 18381	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Poset ∧ 𝑍 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑍 < 𝑌 ↔ (𝑍 ≤ 𝑌 ∧ 𝑍 ≠ 𝑌)))
13	8, 9, 10, 12	syl3anc 1398	. . . . . . . . 9 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑍 < 𝑌 ↔ (𝑍 ≤ 𝑌 ∧ 𝑍 ≠ 𝑌)))
14		df-ne 2959	. . . . . . . . . 10 ⊢ (𝑍 ≠ 𝑌 ↔ ¬ 𝑍 = 𝑌)
15	14	anbi2i 634	. . . . . . . . 9 ⊢ ((𝑍 ≤ 𝑌 ∧ 𝑍 ≠ 𝑌) ↔ (𝑍 ≤ 𝑌 ∧ ¬ 𝑍 = 𝑌))
16	13, 15	bitrdi 290	. . . . . . . 8 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑍 < 𝑌 ↔ (𝑍 ≤ 𝑌 ∧ ¬ 𝑍 = 𝑌)))
17	16	anbi2d 641	. . . . . . 7 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 < 𝑍 ∧ 𝑍 < 𝑌) ↔ (𝑋 < 𝑍 ∧ (𝑍 ≤ 𝑌 ∧ ¬ 𝑍 = 𝑌))))
18	7, 17	bitr4id 293	. . . . . 6 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (((𝑋 < 𝑍 ∧ 𝑍 ≤ 𝑌) ∧ ¬ 𝑍 = 𝑌) ↔ (𝑋 < 𝑍 ∧ 𝑍 < 𝑌)))
19	18	notbid 321	. . . . 5 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (¬ ((𝑋 < 𝑍 ∧ 𝑍 ≤ 𝑌) ∧ ¬ 𝑍 = 𝑌) ↔ ¬ (𝑋 < 𝑍 ∧ 𝑍 < 𝑌)))
20	6, 19	bitr2id 287	. . . 4 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (¬ (𝑋 < 𝑍 ∧ 𝑍 < 𝑌) ↔ ((𝑋 < 𝑍 ∧ 𝑍 ≤ 𝑌) → 𝑍 = 𝑌)))
21	5, 20	sylibd 242	. . 3 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋𝐶𝑌 → ((𝑋 < 𝑍 ∧ 𝑍 ≤ 𝑌) → 𝑍 = 𝑌)))
22	21	3impia 1135	. 2 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → ((𝑋 < 𝑍 ∧ 𝑍 ≤ 𝑌) → 𝑍 = 𝑌))
23	1, 2, 3	cvrlt 40044	. . . . . . 7 ⊢ (((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → 𝑋 < 𝑌)
24	23	ex 417	. . . . . 6 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐶𝑌 → 𝑋 < 𝑌))
25	24	3adant3r3 1203	. . . . 5 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋𝐶𝑌 → 𝑋 < 𝑌))
26	25	3impia 1135	. . . 4 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → 𝑋 < 𝑌)
27		breq2 5113	. . . 4 ⊢ (𝑍 = 𝑌 → (𝑋 < 𝑍 ↔ 𝑋 < 𝑌))
28	26, 27	syl5ibrcom 250	. . 3 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → (𝑍 = 𝑌 → 𝑋 < 𝑍))
29	1, 11	posref 18369	. . . . . 6 ⊢ ((𝐾 ∈ Poset ∧ 𝑌 ∈ 𝐵) → 𝑌 ≤ 𝑌)
30	29	3ad2antr2 1208	. . . . 5 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝑌 ≤ 𝑌)
31		breq1 5112	. . . . 5 ⊢ (𝑍 = 𝑌 → (𝑍 ≤ 𝑌 ↔ 𝑌 ≤ 𝑌))
32	30, 31	syl5ibrcom 250	. . . 4 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑍 = 𝑌 → 𝑍 ≤ 𝑌))
33	32	3adant3 1150	. . 3 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → (𝑍 = 𝑌 → 𝑍 ≤ 𝑌))
34	28, 33	jcad 521	. 2 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → (𝑍 = 𝑌 → (𝑋 < 𝑍 ∧ 𝑍 ≤ 𝑌)))
35	22, 34	impbid 215	1 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → ((𝑋 < 𝑍 ∧ 𝑍 ≤ 𝑌) ↔ 𝑍 = 𝑌))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∧ 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:  cvrval3  40187  cvrexchlem  40193