Theorem cvrnbtwn3 39276

Description: The covers relation implies no in-betweenness. (cvnbtwn3 32224 analog.) (Contributed by NM, 4-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
cvrnbtwn3	⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → ((𝑋 ≤ 𝑍 ∧ 𝑍 < 𝑌) ↔ 𝑋 = 𝑍))

Proof of Theorem cvrnbtwn3

Step	Hyp	Ref	Expression
1		cvrletr.b	. . . 4 ⊢ 𝐵 = (Base‘𝐾)
2		cvrletr.s	. . . 4 ⊢ < = (lt‘𝐾)
3		cvrletr.c	. . . 4 ⊢ 𝐶 = ( ⋖ ‘𝐾)
4	1, 2, 3	cvrnbtwn 39271	. . 3 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → ¬ (𝑋 < 𝑍 ∧ 𝑍 < 𝑌))
5		cvrletr.l	. . . . . . . . 9 ⊢ ≤ = (le‘𝐾)
6	5, 2	pltval 18298	. . . . . . . 8 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (𝑋 < 𝑍 ↔ (𝑋 ≤ 𝑍 ∧ 𝑋 ≠ 𝑍)))
7	6	3adant3r2 1184	. . . . . . 7 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 < 𝑍 ↔ (𝑋 ≤ 𝑍 ∧ 𝑋 ≠ 𝑍)))
8	7	3adant3 1132	. . . . . 6 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → (𝑋 < 𝑍 ↔ (𝑋 ≤ 𝑍 ∧ 𝑋 ≠ 𝑍)))
9	8	anbi1d 631	. . . . 5 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → ((𝑋 < 𝑍 ∧ 𝑍 < 𝑌) ↔ ((𝑋 ≤ 𝑍 ∧ 𝑋 ≠ 𝑍) ∧ 𝑍 < 𝑌)))
10	9	notbid 318	. . . 4 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → (¬ (𝑋 < 𝑍 ∧ 𝑍 < 𝑌) ↔ ¬ ((𝑋 ≤ 𝑍 ∧ 𝑋 ≠ 𝑍) ∧ 𝑍 < 𝑌)))
11		an32 646	. . . . . . 7 ⊢ (((𝑋 ≤ 𝑍 ∧ 𝑋 ≠ 𝑍) ∧ 𝑍 < 𝑌) ↔ ((𝑋 ≤ 𝑍 ∧ 𝑍 < 𝑌) ∧ 𝑋 ≠ 𝑍))
12		df-ne 2927	. . . . . . . 8 ⊢ (𝑋 ≠ 𝑍 ↔ ¬ 𝑋 = 𝑍)
13	12	anbi2i 623	. . . . . . 7 ⊢ (((𝑋 ≤ 𝑍 ∧ 𝑍 < 𝑌) ∧ 𝑋 ≠ 𝑍) ↔ ((𝑋 ≤ 𝑍 ∧ 𝑍 < 𝑌) ∧ ¬ 𝑋 = 𝑍))
14	11, 13	bitri 275	. . . . . 6 ⊢ (((𝑋 ≤ 𝑍 ∧ 𝑋 ≠ 𝑍) ∧ 𝑍 < 𝑌) ↔ ((𝑋 ≤ 𝑍 ∧ 𝑍 < 𝑌) ∧ ¬ 𝑋 = 𝑍))
15	14	notbii 320	. . . . 5 ⊢ (¬ ((𝑋 ≤ 𝑍 ∧ 𝑋 ≠ 𝑍) ∧ 𝑍 < 𝑌) ↔ ¬ ((𝑋 ≤ 𝑍 ∧ 𝑍 < 𝑌) ∧ ¬ 𝑋 = 𝑍))
16		iman 401	. . . . 5 ⊢ (((𝑋 ≤ 𝑍 ∧ 𝑍 < 𝑌) → 𝑋 = 𝑍) ↔ ¬ ((𝑋 ≤ 𝑍 ∧ 𝑍 < 𝑌) ∧ ¬ 𝑋 = 𝑍))
17	15, 16	bitr4i 278	. . . 4 ⊢ (¬ ((𝑋 ≤ 𝑍 ∧ 𝑋 ≠ 𝑍) ∧ 𝑍 < 𝑌) ↔ ((𝑋 ≤ 𝑍 ∧ 𝑍 < 𝑌) → 𝑋 = 𝑍))
18	10, 17	bitrdi 287	. . 3 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → (¬ (𝑋 < 𝑍 ∧ 𝑍 < 𝑌) ↔ ((𝑋 ≤ 𝑍 ∧ 𝑍 < 𝑌) → 𝑋 = 𝑍)))
19	4, 18	mpbid 232	. 2 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → ((𝑋 ≤ 𝑍 ∧ 𝑍 < 𝑌) → 𝑋 = 𝑍))
20	1, 5	posref 18286	. . . . . 6 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵) → 𝑋 ≤ 𝑋)
21		breq2 5114	. . . . . 6 ⊢ (𝑋 = 𝑍 → (𝑋 ≤ 𝑋 ↔ 𝑋 ≤ 𝑍))
22	20, 21	syl5ibcom 245	. . . . 5 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵) → (𝑋 = 𝑍 → 𝑋 ≤ 𝑍))
23	22	3ad2antr1 1189	. . . 4 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 = 𝑍 → 𝑋 ≤ 𝑍))
24	23	3adant3 1132	. . 3 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → (𝑋 = 𝑍 → 𝑋 ≤ 𝑍))
25		simp1 1136	. . . . 5 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → 𝐾 ∈ Poset)
26		simp21 1207	. . . . 5 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → 𝑋 ∈ 𝐵)
27		simp22 1208	. . . . 5 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → 𝑌 ∈ 𝐵)
28		simp3 1138	. . . . 5 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → 𝑋𝐶𝑌)
29	1, 2, 3	cvrlt 39270	. . . . 5 ⊢ (((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → 𝑋 < 𝑌)
30	25, 26, 27, 28, 29	syl31anc 1375	. . . 4 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → 𝑋 < 𝑌)
31		breq1 5113	. . . 4 ⊢ (𝑋 = 𝑍 → (𝑋 < 𝑌 ↔ 𝑍 < 𝑌))
32	30, 31	syl5ibcom 245	. . 3 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → (𝑋 = 𝑍 → 𝑍 < 𝑌))
33	24, 32	jcad 512	. 2 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → (𝑋 = 𝑍 → (𝑋 ≤ 𝑍 ∧ 𝑍 < 𝑌)))
34	19, 33	impbid 212	1 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → ((𝑋 ≤ 𝑍 ∧ 𝑍 < 𝑌) ↔ 𝑋 = 𝑍))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2926   class class class wbr 5110  ‘cfv 6514  Basecbs 17186  lecple 17234  Posetcpo 18275  ltcplt 18276   ⋖ ccvr 39262

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3757  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-iota 6467  df-fun 6516  df-fv 6522  df-proset 18262  df-poset 18281  df-plt 18296  df-covers 39266

This theorem is referenced by:  atcvreq0  39314  cvratlem  39422