Theorem cvrnbtwn3 40334

Description: The covers relation implies no in-betweenness. (cvnbtwn3 32890 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 40329	. . 3 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → ¬ (𝑋 < 𝑍 ∧ 𝑍 < 𝑌))
5		cvrletr.l	. . . . . . . . 9 ⊢ ≤ = (le‘𝐾)
6	5, 2	pltval 18504	. . . . . . . 8 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (𝑋 < 𝑍 ↔ (𝑋 ≤ 𝑍 ∧ 𝑋 ≠ 𝑍)))
7	6	3adant3r2 1202	. . . . . . 7 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 < 𝑍 ↔ (𝑋 ≤ 𝑍 ∧ 𝑋 ≠ 𝑍)))
8	7	3adant3 1150	. . . . . 6 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → (𝑋 < 𝑍 ↔ (𝑋 ≤ 𝑍 ∧ 𝑋 ≠ 𝑍)))
9	8	anbi1d 643	. . . . 5 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → ((𝑋 < 𝑍 ∧ 𝑍 < 𝑌) ↔ ((𝑋 ≤ 𝑍 ∧ 𝑋 ≠ 𝑍) ∧ 𝑍 < 𝑌)))
10	9	notbid 321	. . . 4 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → (¬ (𝑋 < 𝑍 ∧ 𝑍 < 𝑌) ↔ ¬ ((𝑋 ≤ 𝑍 ∧ 𝑋 ≠ 𝑍) ∧ 𝑍 < 𝑌)))
11		an32 659	. . . . . . 7 ⊢ (((𝑋 ≤ 𝑍 ∧ 𝑋 ≠ 𝑍) ∧ 𝑍 < 𝑌) ↔ ((𝑋 ≤ 𝑍 ∧ 𝑍 < 𝑌) ∧ 𝑋 ≠ 𝑍))
12		df-ne 2956	. . . . . . . 8 ⊢ (𝑋 ≠ 𝑍 ↔ ¬ 𝑋 = 𝑍)
13	12	anbi2i 635	. . . . . . 7 ⊢ (((𝑋 ≤ 𝑍 ∧ 𝑍 < 𝑌) ∧ 𝑋 ≠ 𝑍) ↔ ((𝑋 ≤ 𝑍 ∧ 𝑍 < 𝑌) ∧ ¬ 𝑋 = 𝑍))
14	11, 13	bitri 278	. . . . . 6 ⊢ (((𝑋 ≤ 𝑍 ∧ 𝑋 ≠ 𝑍) ∧ 𝑍 < 𝑌) ↔ ((𝑋 ≤ 𝑍 ∧ 𝑍 < 𝑌) ∧ ¬ 𝑋 = 𝑍))
15	14	notbii 323	. . . . 5 ⊢ (¬ ((𝑋 ≤ 𝑍 ∧ 𝑋 ≠ 𝑍) ∧ 𝑍 < 𝑌) ↔ ¬ ((𝑋 ≤ 𝑍 ∧ 𝑍 < 𝑌) ∧ ¬ 𝑋 = 𝑍))
16		iman 407	. . . . 5 ⊢ (((𝑋 ≤ 𝑍 ∧ 𝑍 < 𝑌) → 𝑋 = 𝑍) ↔ ¬ ((𝑋 ≤ 𝑍 ∧ 𝑍 < 𝑌) ∧ ¬ 𝑋 = 𝑍))
17	15, 16	bitr4i 281	. . . 4 ⊢ (¬ ((𝑋 ≤ 𝑍 ∧ 𝑋 ≠ 𝑍) ∧ 𝑍 < 𝑌) ↔ ((𝑋 ≤ 𝑍 ∧ 𝑍 < 𝑌) → 𝑋 = 𝑍))
18	10, 17	bitrdi 290	. . 3 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → (¬ (𝑋 < 𝑍 ∧ 𝑍 < 𝑌) ↔ ((𝑋 ≤ 𝑍 ∧ 𝑍 < 𝑌) → 𝑋 = 𝑍)))
19	4, 18	mpbid 235	. 2 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → ((𝑋 ≤ 𝑍 ∧ 𝑍 < 𝑌) → 𝑋 = 𝑍))
20	1, 5	posref 18492	. . . . . 6 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵) → 𝑋 ≤ 𝑋)
21		breq2 5106	. . . . . 6 ⊢ (𝑋 = 𝑍 → (𝑋 ≤ 𝑋 ↔ 𝑋 ≤ 𝑍))
22	20, 21	syl5ibcom 248	. . . . 5 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵) → (𝑋 = 𝑍 → 𝑋 ≤ 𝑍))
23	22	3ad2antr1 1207	. . . 4 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 = 𝑍 → 𝑋 ≤ 𝑍))
24	23	3adant3 1150	. . 3 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → (𝑋 = 𝑍 → 𝑋 ≤ 𝑍))
25		simp1 1154	. . . . 5 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → 𝐾 ∈ Poset)
26		simp21 1225	. . . . 5 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → 𝑋 ∈ 𝐵)
27		simp22 1226	. . . . 5 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → 𝑌 ∈ 𝐵)
28		simp3 1156	. . . . 5 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → 𝑋𝐶𝑌)
29	1, 2, 3	cvrlt 40328	. . . . 5 ⊢ (((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → 𝑋 < 𝑌)
30	25, 26, 27, 28, 29	syl31anc 1400	. . . 4 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → 𝑋 < 𝑌)
31		breq1 5105	. . . 4 ⊢ (𝑋 = 𝑍 → (𝑋 < 𝑌 ↔ 𝑍 < 𝑌))
32	30, 31	syl5ibcom 248	. . 3 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → (𝑋 = 𝑍 → 𝑍 < 𝑌))
33	24, 32	jcad 522	. 2 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → (𝑋 = 𝑍 → (𝑋 ≤ 𝑍 ∧ 𝑍 < 𝑌)))
34	19, 33	impbid 215	1 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → ((𝑋 ≤ 𝑍 ∧ 𝑍 < 𝑌) ↔ 𝑋 = 𝑍))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144   ≠ wne 2955   class class class wbr 5102  ‘cfv 6538  Basecbs 17387  lecple 17435  Posetcpo 18481  ltcplt 18482   ⋖ ccvr 40320

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-iota 6494  df-fun 6540  df-fv 6546  df-proset 18468  df-poset 18487  df-plt 18502  df-covers 40324

This theorem is used by:  atcvreq0  40372  cvratlem  40479