Theorem cvrnbtwn3 33581

Description: The covers relation implies no in-betweenness. (cvnbtwn3 28531 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 33576	. . 3 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → ¬ (𝑋 < 𝑍 ∧ 𝑍 < 𝑌))
5		cvrletr.l	. . . . . . . . 9 ⊢ ≤ = (le‘𝐾)
6	5, 2	pltval 16783	. . . . . . . 8 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (𝑋 < 𝑍 ↔ (𝑋 ≤ 𝑍 ∧ 𝑋 ≠ 𝑍)))
7	6	3adant3r2 1267	. . . . . . 7 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 < 𝑍 ↔ (𝑋 ≤ 𝑍 ∧ 𝑋 ≠ 𝑍)))
8	7	3adant3 1074	. . . . . 6 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → (𝑋 < 𝑍 ↔ (𝑋 ≤ 𝑍 ∧ 𝑋 ≠ 𝑍)))
9	8	anbi1d 737	. . . . 5 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → ((𝑋 < 𝑍 ∧ 𝑍 < 𝑌) ↔ ((𝑋 ≤ 𝑍 ∧ 𝑋 ≠ 𝑍) ∧ 𝑍 < 𝑌)))
10	9	notbid 307	. . . 4 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → (¬ (𝑋 < 𝑍 ∧ 𝑍 < 𝑌) ↔ ¬ ((𝑋 ≤ 𝑍 ∧ 𝑋 ≠ 𝑍) ∧ 𝑍 < 𝑌)))
11		an32 835	. . . . . . 7 ⊢ (((𝑋 ≤ 𝑍 ∧ 𝑋 ≠ 𝑍) ∧ 𝑍 < 𝑌) ↔ ((𝑋 ≤ 𝑍 ∧ 𝑍 < 𝑌) ∧ 𝑋 ≠ 𝑍))
12		df-ne 2782	. . . . . . . 8 ⊢ (𝑋 ≠ 𝑍 ↔ ¬ 𝑋 = 𝑍)
13	12	anbi2i 726	. . . . . . 7 ⊢ (((𝑋 ≤ 𝑍 ∧ 𝑍 < 𝑌) ∧ 𝑋 ≠ 𝑍) ↔ ((𝑋 ≤ 𝑍 ∧ 𝑍 < 𝑌) ∧ ¬ 𝑋 = 𝑍))
14	11, 13	bitri 263	. . . . . 6 ⊢ (((𝑋 ≤ 𝑍 ∧ 𝑋 ≠ 𝑍) ∧ 𝑍 < 𝑌) ↔ ((𝑋 ≤ 𝑍 ∧ 𝑍 < 𝑌) ∧ ¬ 𝑋 = 𝑍))
15	14	notbii 309	. . . . 5 ⊢ (¬ ((𝑋 ≤ 𝑍 ∧ 𝑋 ≠ 𝑍) ∧ 𝑍 < 𝑌) ↔ ¬ ((𝑋 ≤ 𝑍 ∧ 𝑍 < 𝑌) ∧ ¬ 𝑋 = 𝑍))
16		iman 439	. . . . 5 ⊢ (((𝑋 ≤ 𝑍 ∧ 𝑍 < 𝑌) → 𝑋 = 𝑍) ↔ ¬ ((𝑋 ≤ 𝑍 ∧ 𝑍 < 𝑌) ∧ ¬ 𝑋 = 𝑍))
17	15, 16	bitr4i 266	. . . 4 ⊢ (¬ ((𝑋 ≤ 𝑍 ∧ 𝑋 ≠ 𝑍) ∧ 𝑍 < 𝑌) ↔ ((𝑋 ≤ 𝑍 ∧ 𝑍 < 𝑌) → 𝑋 = 𝑍))
18	10, 17	syl6bb 275	. . 3 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → (¬ (𝑋 < 𝑍 ∧ 𝑍 < 𝑌) ↔ ((𝑋 ≤ 𝑍 ∧ 𝑍 < 𝑌) → 𝑋 = 𝑍)))
19	4, 18	mpbid 221	. 2 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → ((𝑋 ≤ 𝑍 ∧ 𝑍 < 𝑌) → 𝑋 = 𝑍))
20	1, 5	posref 16774	. . . . . 6 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵) → 𝑋 ≤ 𝑋)
21		breq2 4587	. . . . . 6 ⊢ (𝑋 = 𝑍 → (𝑋 ≤ 𝑋 ↔ 𝑋 ≤ 𝑍))
22	20, 21	syl5ibcom 234	. . . . 5 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵) → (𝑋 = 𝑍 → 𝑋 ≤ 𝑍))
23	22	3ad2antr1 1219	. . . 4 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 = 𝑍 → 𝑋 ≤ 𝑍))
24	23	3adant3 1074	. . 3 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → (𝑋 = 𝑍 → 𝑋 ≤ 𝑍))
25		simp1 1054	. . . . 5 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → 𝐾 ∈ Poset)
26		simp21 1087	. . . . 5 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → 𝑋 ∈ 𝐵)
27		simp22 1088	. . . . 5 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → 𝑌 ∈ 𝐵)
28		simp3 1056	. . . . 5 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → 𝑋𝐶𝑌)
29	1, 2, 3	cvrlt 33575	. . . . 5 ⊢ (((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → 𝑋 < 𝑌)
30	25, 26, 27, 28, 29	syl31anc 1321	. . . 4 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → 𝑋 < 𝑌)
31		breq1 4586	. . . 4 ⊢ (𝑋 = 𝑍 → (𝑋 < 𝑌 ↔ 𝑍 < 𝑌))
32	30, 31	syl5ibcom 234	. . 3 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → (𝑋 = 𝑍 → 𝑍 < 𝑌))
33	24, 32	jcad 554	. 2 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → (𝑋 = 𝑍 → (𝑋 ≤ 𝑍 ∧ 𝑍 < 𝑌)))
34	19, 33	impbid 201	1 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → ((𝑋 ≤ 𝑍 ∧ 𝑍 < 𝑌) ↔ 𝑋 = 𝑍))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780   class class class wbr 4583  ‘cfv 5804  Basecbs 15695  lecple 15775  Posetcpo 16763  ltcplt 16764   ⋖ ccvr 33567

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-preset 16751  df-poset 16769  df-plt 16781  df-covers 33571

This theorem is referenced by:  atcvreq0  33619  cvratlem  33725