Theorem cvrnbtwn2 39767

Description: The covers relation implies no in-betweenness. (cvnbtwn2 32376 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 39763	. . . . 5 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → ¬ (𝑋 < 𝑍 ∧ 𝑍 < 𝑌))
5	4	3expia 1127	. . . 4 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋𝐶𝑌 → ¬ (𝑋 < 𝑍 ∧ 𝑍 < 𝑌)))
6		iman 402	. . . . 5 ⊢ (((𝑋 < 𝑍 ∧ 𝑍 ≤ 𝑌) → 𝑍 = 𝑌) ↔ ¬ ((𝑋 < 𝑍 ∧ 𝑍 ≤ 𝑌) ∧ ¬ 𝑍 = 𝑌))
7		anass 469	. . . . . . 7 ⊢ (((𝑋 < 𝑍 ∧ 𝑍 ≤ 𝑌) ∧ ¬ 𝑍 = 𝑌) ↔ (𝑋 < 𝑍 ∧ (𝑍 ≤ 𝑌 ∧ ¬ 𝑍 = 𝑌)))
8		simpl 483	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝐾 ∈ Poset)
9		simpr3 1203	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝑍 ∈ 𝐵)
10		simpr2 1202	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝑌 ∈ 𝐵)
11		cvrletr.l	. . . . . . . . . . 11 ⊢ ≤ = (le‘𝐾)
12	11, 2	pltval 18287	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Poset ∧ 𝑍 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑍 < 𝑌 ↔ (𝑍 ≤ 𝑌 ∧ 𝑍 ≠ 𝑌)))
13	8, 9, 10, 12	syl3anc 1379	. . . . . . . . 9 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑍 < 𝑌 ↔ (𝑍 ≤ 𝑌 ∧ 𝑍 ≠ 𝑌)))
14		df-ne 2935	. . . . . . . . . 10 ⊢ (𝑍 ≠ 𝑌 ↔ ¬ 𝑍 = 𝑌)
15	14	anbi2i 629	. . . . . . . . 9 ⊢ ((𝑍 ≤ 𝑌 ∧ 𝑍 ≠ 𝑌) ↔ (𝑍 ≤ 𝑌 ∧ ¬ 𝑍 = 𝑌))
16	13, 15	bitrdi 288	. . . . . . . 8 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑍 < 𝑌 ↔ (𝑍 ≤ 𝑌 ∧ ¬ 𝑍 = 𝑌)))
17	16	anbi2d 636	. . . . . . 7 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 < 𝑍 ∧ 𝑍 < 𝑌) ↔ (𝑋 < 𝑍 ∧ (𝑍 ≤ 𝑌 ∧ ¬ 𝑍 = 𝑌))))
18	7, 17	bitr4id 291	. . . . . 6 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (((𝑋 < 𝑍 ∧ 𝑍 ≤ 𝑌) ∧ ¬ 𝑍 = 𝑌) ↔ (𝑋 < 𝑍 ∧ 𝑍 < 𝑌)))
19	18	notbid 319	. . . . 5 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (¬ ((𝑋 < 𝑍 ∧ 𝑍 ≤ 𝑌) ∧ ¬ 𝑍 = 𝑌) ↔ ¬ (𝑋 < 𝑍 ∧ 𝑍 < 𝑌)))
20	6, 19	bitr2id 285	. . . 4 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (¬ (𝑋 < 𝑍 ∧ 𝑍 < 𝑌) ↔ ((𝑋 < 𝑍 ∧ 𝑍 ≤ 𝑌) → 𝑍 = 𝑌)))
21	5, 20	sylibd 240	. . 3 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋𝐶𝑌 → ((𝑋 < 𝑍 ∧ 𝑍 ≤ 𝑌) → 𝑍 = 𝑌)))
22	21	3impia 1123	. 2 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → ((𝑋 < 𝑍 ∧ 𝑍 ≤ 𝑌) → 𝑍 = 𝑌))
23	1, 2, 3	cvrlt 39762	. . . . . . 7 ⊢ (((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → 𝑋 < 𝑌)
24	23	ex 413	. . . . . 6 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐶𝑌 → 𝑋 < 𝑌))
25	24	3adant3r3 1191	. . . . 5 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋𝐶𝑌 → 𝑋 < 𝑌))
26	25	3impia 1123	. . . 4 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → 𝑋 < 𝑌)
27		breq2 5076	. . . 4 ⊢ (𝑍 = 𝑌 → (𝑋 < 𝑍 ↔ 𝑋 < 𝑌))
28	26, 27	syl5ibrcom 248	. . 3 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → (𝑍 = 𝑌 → 𝑋 < 𝑍))
29	1, 11	posref 18275	. . . . . 6 ⊢ ((𝐾 ∈ Poset ∧ 𝑌 ∈ 𝐵) → 𝑌 ≤ 𝑌)
30	29	3ad2antr2 1196	. . . . 5 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝑌 ≤ 𝑌)
31		breq1 5075	. . . . 5 ⊢ (𝑍 = 𝑌 → (𝑍 ≤ 𝑌 ↔ 𝑌 ≤ 𝑌))
32	30, 31	syl5ibrcom 248	. . . 4 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑍 = 𝑌 → 𝑍 ≤ 𝑌))
33	32	3adant3 1138	. . 3 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → (𝑍 = 𝑌 → 𝑍 ≤ 𝑌))
34	28, 33	jcad 517	. 2 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → (𝑍 = 𝑌 → (𝑋 < 𝑍 ∧ 𝑍 ≤ 𝑌)))
35	22, 34	impbid 213	1 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → ((𝑋 < 𝑍 ∧ 𝑍 ≤ 𝑌) ↔ 𝑍 = 𝑌))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119   ≠ wne 2934   class class class wbr 5072  ‘cfv 6485  Basecbs 17170  lecple 17218  Posetcpo 18264  ltcplt 18265   ⋖ ccvr 39754

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-sbc 3724  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-iota 6441  df-fun 6487  df-fv 6493  df-proset 18251  df-poset 18270  df-plt 18285  df-covers 39758

This theorem is referenced by:  cvrval3  39905  cvrexchlem  39911