Theorem cvrnbtwn3 36406

Description: The covers relation implies no in-betweenness. (cvnbtwn3 30059 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 36401	. . 3 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → ¬ (𝑋 < 𝑍 ∧ 𝑍 < 𝑌))
5		cvrletr.l	. . . . . . . . 9 ⊢ ≤ = (le‘𝐾)
6	5, 2	pltval 17564	. . . . . . . 8 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (𝑋 < 𝑍 ↔ (𝑋 ≤ 𝑍 ∧ 𝑋 ≠ 𝑍)))
7	6	3adant3r2 1179	. . . . . . 7 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 < 𝑍 ↔ (𝑋 ≤ 𝑍 ∧ 𝑋 ≠ 𝑍)))
8	7	3adant3 1128	. . . . . 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 320	. . . 4 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → (¬ (𝑋 < 𝑍 ∧ 𝑍 < 𝑌) ↔ ¬ ((𝑋 ≤ 𝑍 ∧ 𝑋 ≠ 𝑍) ∧ 𝑍 < 𝑌)))
11		an32 644	. . . . . . 7 ⊢ (((𝑋 ≤ 𝑍 ∧ 𝑋 ≠ 𝑍) ∧ 𝑍 < 𝑌) ↔ ((𝑋 ≤ 𝑍 ∧ 𝑍 < 𝑌) ∧ 𝑋 ≠ 𝑍))
12		df-ne 3017	. . . . . . . 8 ⊢ (𝑋 ≠ 𝑍 ↔ ¬ 𝑋 = 𝑍)
13	12	anbi2i 624	. . . . . . 7 ⊢ (((𝑋 ≤ 𝑍 ∧ 𝑍 < 𝑌) ∧ 𝑋 ≠ 𝑍) ↔ ((𝑋 ≤ 𝑍 ∧ 𝑍 < 𝑌) ∧ ¬ 𝑋 = 𝑍))
14	11, 13	bitri 277	. . . . . 6 ⊢ (((𝑋 ≤ 𝑍 ∧ 𝑋 ≠ 𝑍) ∧ 𝑍 < 𝑌) ↔ ((𝑋 ≤ 𝑍 ∧ 𝑍 < 𝑌) ∧ ¬ 𝑋 = 𝑍))
15	14	notbii 322	. . . . 5 ⊢ (¬ ((𝑋 ≤ 𝑍 ∧ 𝑋 ≠ 𝑍) ∧ 𝑍 < 𝑌) ↔ ¬ ((𝑋 ≤ 𝑍 ∧ 𝑍 < 𝑌) ∧ ¬ 𝑋 = 𝑍))
16		iman 404	. . . . 5 ⊢ (((𝑋 ≤ 𝑍 ∧ 𝑍 < 𝑌) → 𝑋 = 𝑍) ↔ ¬ ((𝑋 ≤ 𝑍 ∧ 𝑍 < 𝑌) ∧ ¬ 𝑋 = 𝑍))
17	15, 16	bitr4i 280	. . . 4 ⊢ (¬ ((𝑋 ≤ 𝑍 ∧ 𝑋 ≠ 𝑍) ∧ 𝑍 < 𝑌) ↔ ((𝑋 ≤ 𝑍 ∧ 𝑍 < 𝑌) → 𝑋 = 𝑍))
18	10, 17	syl6bb 289	. . 3 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → (¬ (𝑋 < 𝑍 ∧ 𝑍 < 𝑌) ↔ ((𝑋 ≤ 𝑍 ∧ 𝑍 < 𝑌) → 𝑋 = 𝑍)))
19	4, 18	mpbid 234	. 2 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → ((𝑋 ≤ 𝑍 ∧ 𝑍 < 𝑌) → 𝑋 = 𝑍))
20	1, 5	posref 17555	. . . . . 6 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵) → 𝑋 ≤ 𝑋)
21		breq2 5062	. . . . . 6 ⊢ (𝑋 = 𝑍 → (𝑋 ≤ 𝑋 ↔ 𝑋 ≤ 𝑍))
22	20, 21	syl5ibcom 247	. . . . 5 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵) → (𝑋 = 𝑍 → 𝑋 ≤ 𝑍))
23	22	3ad2antr1 1184	. . . 4 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 = 𝑍 → 𝑋 ≤ 𝑍))
24	23	3adant3 1128	. . 3 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → (𝑋 = 𝑍 → 𝑋 ≤ 𝑍))
25		simp1 1132	. . . . 5 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → 𝐾 ∈ Poset)
26		simp21 1202	. . . . 5 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → 𝑋 ∈ 𝐵)
27		simp22 1203	. . . . 5 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → 𝑌 ∈ 𝐵)
28		simp3 1134	. . . . 5 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → 𝑋𝐶𝑌)
29	1, 2, 3	cvrlt 36400	. . . . 5 ⊢ (((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → 𝑋 < 𝑌)
30	25, 26, 27, 28, 29	syl31anc 1369	. . . 4 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → 𝑋 < 𝑌)
31		breq1 5061	. . . 4 ⊢ (𝑋 = 𝑍 → (𝑋 < 𝑌 ↔ 𝑍 < 𝑌))
32	30, 31	syl5ibcom 247	. . 3 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → (𝑋 = 𝑍 → 𝑍 < 𝑌))
33	24, 32	jcad 515	. 2 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → (𝑋 = 𝑍 → (𝑋 ≤ 𝑍 ∧ 𝑍 < 𝑌)))
34	19, 33	impbid 214	1 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → ((𝑋 ≤ 𝑍 ∧ 𝑍 < 𝑌) ↔ 𝑋 = 𝑍))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1533   ∈ wcel 2110   ≠ wne 3016   class class class wbr 5058  ‘cfv 6349  Basecbs 16477  lecple 16566  Posetcpo 17544  ltcplt 17545   ⋖ ccvr 36392

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-iota 6308  df-fun 6351  df-fv 6357  df-proset 17532  df-poset 17550  df-plt 17562  df-covers 36396

This theorem is referenced by:  atcvreq0  36444  cvratlem  36551