Theorem cvrnbtwn2 40332

Description: The covers relation implies no in-betweenness. (cvnbtwn2 32889 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 40328	. . . . 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 407	. . . . 5 ⊢ (((𝑋 < 𝑍 ∧ 𝑍 ≤ 𝑌) → 𝑍 = 𝑌) ↔ ¬ ((𝑋 < 𝑍 ∧ 𝑍 ≤ 𝑌) ∧ ¬ 𝑍 = 𝑌))
7		anass 474	. . . . . . 7 ⊢ (((𝑋 < 𝑍 ∧ 𝑍 ≤ 𝑌) ∧ ¬ 𝑍 = 𝑌) ↔ (𝑋 < 𝑍 ∧ (𝑍 ≤ 𝑌 ∧ ¬ 𝑍 = 𝑌)))
8		simpl 488	. . . . . . . . . 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 18504	. . . . . . . . . 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 2957	. . . . . . . . . 10 ⊢ (𝑍 ≠ 𝑌 ↔ ¬ 𝑍 = 𝑌)
15	14	anbi2i 635	. . . . . . . . 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 642	. . . . . . 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 40327	. . . . . . 7 ⊢ (((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → 𝑋 < 𝑌)
24	23	ex 418	. . . . . 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 5107	. . . 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 18492	. . . . . 6 ⊢ ((𝐾 ∈ Poset ∧ 𝑌 ∈ 𝐵) → 𝑌 ≤ 𝑌)
30	29	3ad2antr2 1208	. . . . 5 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝑌 ≤ 𝑌)
31		breq1 5106	. . . . 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 522	. 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))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ≠ wne 2956   class class class wbr 5103  ‘cfv 6538  Basecbs 17387  lecple 17435  Posetcpo 18481  ltcplt 18482   ⋖ ccvr 40319

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-sbc 3740  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  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-proset 18468  df-poset 18487  df-plt 18502  df-covers 40323

This theorem is used by:  cvrval3  40470  cvrexchlem  40476