Theorem cvrnbtwn3 40078

Description: The covers relation implies no in-betweenness. (cvnbtwn3 32651 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 40073	. . 3 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → ¬ (𝑋 < 𝑍 ∧ 𝑍 < 𝑌))
5		cvrletr.l	. . . . . . . . 9 ⊢ ≤ = (le‘𝐾)
6	5, 2	pltval 18392	. . . . . . . 8 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (𝑋 < 𝑍 ↔ (𝑋 ≤ 𝑍 ∧ 𝑋 ≠ 𝑍)))
7	6	3adant3r2 1201	. . . . . . 7 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 < 𝑍 ↔ (𝑋 ≤ 𝑍 ∧ 𝑋 ≠ 𝑍)))
8	7	3adant3 1149	. . . . . 6 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → (𝑋 < 𝑍 ↔ (𝑋 ≤ 𝑍 ∧ 𝑋 ≠ 𝑍)))
9	8	anbi1d 642	. . . . 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 658	. . . . . . 7 ⊢ (((𝑋 ≤ 𝑍 ∧ 𝑋 ≠ 𝑍) ∧ 𝑍 < 𝑌) ↔ ((𝑋 ≤ 𝑍 ∧ 𝑍 < 𝑌) ∧ 𝑋 ≠ 𝑍))
12		df-ne 2958	. . . . . . . 8 ⊢ (𝑋 ≠ 𝑍 ↔ ¬ 𝑋 = 𝑍)
13	12	anbi2i 634	. . . . . . 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 406	. . . . 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 18380	. . . . . 6 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵) → 𝑋 ≤ 𝑋)
21		breq2 5112	. . . . . 6 ⊢ (𝑋 = 𝑍 → (𝑋 ≤ 𝑋 ↔ 𝑋 ≤ 𝑍))
22	20, 21	syl5ibcom 248	. . . . 5 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵) → (𝑋 = 𝑍 → 𝑋 ≤ 𝑍))
23	22	3ad2antr1 1206	. . . 4 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 = 𝑍 → 𝑋 ≤ 𝑍))
24	23	3adant3 1149	. . 3 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → (𝑋 = 𝑍 → 𝑋 ≤ 𝑍))
25		simp1 1153	. . . . 5 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → 𝐾 ∈ Poset)
26		simp21 1224	. . . . 5 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → 𝑋 ∈ 𝐵)
27		simp22 1225	. . . . 5 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → 𝑌 ∈ 𝐵)
28		simp3 1155	. . . . 5 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → 𝑋𝐶𝑌)
29	1, 2, 3	cvrlt 40072	. . . . 5 ⊢ (((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → 𝑋 < 𝑌)
30	25, 26, 27, 28, 29	syl31anc 1399	. . . 4 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → 𝑋 < 𝑌)
31		breq1 5111	. . . 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 521	. 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 400   ∧ w3a 1102   = wceq 1569   ∈ wcel 2142   ≠ wne 2957   class class class wbr 5108  ‘cfv 6536  Basecbs 17275  lecple 17323  Posetcpo 18369  ltcplt 18370   ⋖ ccvr 40064

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5256  ax-nul 5268  ax-pow 5335  ax-pr 5403  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  df-sbc 3744  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5555  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-iota 6492  df-fun 6538  df-fv 6544  df-proset 18356  df-poset 18375  df-plt 18390  df-covers 40068

This theorem is used by:  atcvreq0  40116  cvratlem  40223