Theorem cvrnbtwn2 39738

Description: The covers relation implies no in-betweenness. (cvnbtwn2 32376 analog.) (Contributed by NM, 17-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
cvrnbtwn2	⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → ((𝑋 < 𝑍 ∧ 𝑍 ≤ 𝑌) ↔ 𝑍 = 𝑌))

Proof of Theorem cvrnbtwn2

Step	Hyp	Ref	Expression
1		cvrletr.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐾)
2		cvrletr.s	. . . . . 6 ⊢ < = (lt‘𝐾)
3		cvrletr.c	. . . . . 6 ⊢ 𝐶 = ( ⋖ ‘𝐾)
4	1, 2, 3	cvrnbtwn 39734	. . . . 5 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → ¬ (𝑋 < 𝑍 ∧ 𝑍 < 𝑌))
5	4	3expia 1122	. . . 4 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋𝐶𝑌 → ¬ (𝑋 < 𝑍 ∧ 𝑍 < 𝑌)))
6		iman 401	. . . . 5 ⊢ (((𝑋 < 𝑍 ∧ 𝑍 ≤ 𝑌) → 𝑍 = 𝑌) ↔ ¬ ((𝑋 < 𝑍 ∧ 𝑍 ≤ 𝑌) ∧ ¬ 𝑍 = 𝑌))
7		anass 468	. . . . . . 7 ⊢ (((𝑋 < 𝑍 ∧ 𝑍 ≤ 𝑌) ∧ ¬ 𝑍 = 𝑌) ↔ (𝑋 < 𝑍 ∧ (𝑍 ≤ 𝑌 ∧ ¬ 𝑍 = 𝑌)))
8		simpl 482	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝐾 ∈ Poset)
9		simpr3 1198	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝑍 ∈ 𝐵)
10		simpr2 1197	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝑌 ∈ 𝐵)
11		cvrletr.l	. . . . . . . . . . 11 ⊢ ≤ = (le‘𝐾)
12	11, 2	pltval 18290	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Poset ∧ 𝑍 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑍 < 𝑌 ↔ (𝑍 ≤ 𝑌 ∧ 𝑍 ≠ 𝑌)))
13	8, 9, 10, 12	syl3anc 1374	. . . . . . . . 9 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑍 < 𝑌 ↔ (𝑍 ≤ 𝑌 ∧ 𝑍 ≠ 𝑌)))
14		df-ne 2934	. . . . . . . . . 10 ⊢ (𝑍 ≠ 𝑌 ↔ ¬ 𝑍 = 𝑌)
15	14	anbi2i 624	. . . . . . . . 9 ⊢ ((𝑍 ≤ 𝑌 ∧ 𝑍 ≠ 𝑌) ↔ (𝑍 ≤ 𝑌 ∧ ¬ 𝑍 = 𝑌))
16	13, 15	bitrdi 287	. . . . . . . 8 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑍 < 𝑌 ↔ (𝑍 ≤ 𝑌 ∧ ¬ 𝑍 = 𝑌)))
17	16	anbi2d 631	. . . . . . 7 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 < 𝑍 ∧ 𝑍 < 𝑌) ↔ (𝑋 < 𝑍 ∧ (𝑍 ≤ 𝑌 ∧ ¬ 𝑍 = 𝑌))))
18	7, 17	bitr4id 290	. . . . . 6 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (((𝑋 < 𝑍 ∧ 𝑍 ≤ 𝑌) ∧ ¬ 𝑍 = 𝑌) ↔ (𝑋 < 𝑍 ∧ 𝑍 < 𝑌)))
19	18	notbid 318	. . . . 5 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (¬ ((𝑋 < 𝑍 ∧ 𝑍 ≤ 𝑌) ∧ ¬ 𝑍 = 𝑌) ↔ ¬ (𝑋 < 𝑍 ∧ 𝑍 < 𝑌)))
20	6, 19	bitr2id 284	. . . 4 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (¬ (𝑋 < 𝑍 ∧ 𝑍 < 𝑌) ↔ ((𝑋 < 𝑍 ∧ 𝑍 ≤ 𝑌) → 𝑍 = 𝑌)))
21	5, 20	sylibd 239	. . 3 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋𝐶𝑌 → ((𝑋 < 𝑍 ∧ 𝑍 ≤ 𝑌) → 𝑍 = 𝑌)))
22	21	3impia 1118	. 2 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → ((𝑋 < 𝑍 ∧ 𝑍 ≤ 𝑌) → 𝑍 = 𝑌))
23	1, 2, 3	cvrlt 39733	. . . . . . 7 ⊢ (((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → 𝑋 < 𝑌)
24	23	ex 412	. . . . . 6 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐶𝑌 → 𝑋 < 𝑌))
25	24	3adant3r3 1186	. . . . 5 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋𝐶𝑌 → 𝑋 < 𝑌))
26	25	3impia 1118	. . . 4 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → 𝑋 < 𝑌)
27		breq2 5090	. . . 4 ⊢ (𝑍 = 𝑌 → (𝑋 < 𝑍 ↔ 𝑋 < 𝑌))
28	26, 27	syl5ibrcom 247	. . 3 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → (𝑍 = 𝑌 → 𝑋 < 𝑍))
29	1, 11	posref 18278	. . . . . 6 ⊢ ((𝐾 ∈ Poset ∧ 𝑌 ∈ 𝐵) → 𝑌 ≤ 𝑌)
30	29	3ad2antr2 1191	. . . . 5 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → 𝑌 ≤ 𝑌)
31		breq1 5089	. . . . 5 ⊢ (𝑍 = 𝑌 → (𝑍 ≤ 𝑌 ↔ 𝑌 ≤ 𝑌))
32	30, 31	syl5ibrcom 247	. . . 4 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑍 = 𝑌 → 𝑍 ≤ 𝑌))
33	32	3adant3 1133	. . 3 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → (𝑍 = 𝑌 → 𝑍 ≤ 𝑌))
34	28, 33	jcad 512	. 2 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → (𝑍 = 𝑌 → (𝑋 < 𝑍 ∧ 𝑍 ≤ 𝑌)))
35	22, 34	impbid 212	1 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → ((𝑋 < 𝑍 ∧ 𝑍 ≤ 𝑌) ↔ 𝑍 = 𝑌))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2933   class class class wbr 5086  ‘cfv 6493  Basecbs 17173  lecple 17221  Posetcpo 18267  ltcplt 18268   ⋖ ccvr 39725

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5232  ax-nul 5242  ax-pow 5303  ax-pr 5371  ax-un 7683

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-sbc 3730  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-iota 6449  df-fun 6495  df-fv 6501  df-proset 18254  df-poset 18273  df-plt 18288  df-covers 39729

This theorem is referenced by:  cvrval3  39876  cvrexchlem  39882