Theorem cvnbtwn4t 5721

Description: The covering relation implies no in-betweenness. Part of proof of Lemma 7.5.1 of [MaedaMaeda] p. 31.

Assertion

Ref	Expression
cvnbtwn4t	⊢ ((A ∈ Cℋ ∧ B ∈ Cℋ ∧ C ∈ Cℋ ) → (A ⋖ B → ((A ⊆ C ∧ C ⊆ B) → (C = A ∨ C = B))))

Proof of Theorem cvnbtwn4t

Step	Hyp	Ref	Expression
1		cvnbtwnt 5718	. 2 ⊢ ((A ∈ Cℋ ∧ B ∈ Cℋ ∧ C ∈ Cℋ ) → (A ⋖ B → ¬ (A ⊂ C ∧ C ⊂ B)))
2		iman 205	. . 3 ⊢ (((A ⊆ C ∧ C ⊆ B) → (C = A ∨ C = B)) ↔ ¬ ((A ⊆ C ∧ C ⊆ B) ∧ ¬ (C = A ∨ C = B)))
3		an4 388	. . . . 5 ⊢ (((A ⊆ C ∧ C ⊆ B) ∧ (¬ A = C ∧ ¬ C = B)) ↔ ((A ⊆ C ∧ ¬ A = C) ∧ (C ⊆ B ∧ ¬ C = B)))
4		ioran 254	. . . . . . 7 ⊢ (¬ (C = A ∨ C = B) ↔ (¬ C = A ∧ ¬ C = B))
5		cleqcom 1103	. . . . . . . . 9 ⊢ (C = A ↔ A = C)
6	5	negbii 162	. . . . . . . 8 ⊢ (¬ C = A ↔ ¬ A = C)
7	6	anbi1i 368	. . . . . . 7 ⊢ ((¬ C = A ∧ ¬ C = B) ↔ (¬ A = C ∧ ¬ C = B))
8	4, 7	bitr 151	. . . . . 6 ⊢ (¬ (C = A ∨ C = B) ↔ (¬ A = C ∧ ¬ C = B))
9	8	anbi2i 367	. . . . 5 ⊢ (((A ⊆ C ∧ C ⊆ B) ∧ ¬ (C = A ∨ C = B)) ↔ ((A ⊆ C ∧ C ⊆ B) ∧ (¬ A = C ∧ ¬ C = B)))
10		dfpss2 1557	. . . . . 6 ⊢ (A ⊂ C ↔ (A ⊆ C ∧ ¬ A = C))
11		dfpss2 1557	. . . . . 6 ⊢ (C ⊂ B ↔ (C ⊆ B ∧ ¬ C = B))
12	10, 11	anbi12i 369	. . . . 5 ⊢ ((A ⊂ C ∧ C ⊂ B) ↔ ((A ⊆ C ∧ ¬ A = C) ∧ (C ⊆ B ∧ ¬ C = B)))
13	3, 9, 12	3bitr4 158	. . . 4 ⊢ (((A ⊆ C ∧ C ⊆ B) ∧ ¬ (C = A ∨ C = B)) ↔ (A ⊂ C ∧ C ⊂ B))
14	13	negbii 162	. . 3 ⊢ (¬ ((A ⊆ C ∧ C ⊆ B) ∧ ¬ (C = A ∨ C = B)) ↔ ¬ (A ⊂ C ∧ C ⊂ B))
15	2, 14	bitr2 152	. 2 ⊢ (¬ (A ⊂ C ∧ C ⊂ B) ↔ ((A ⊆ C ∧ C ⊆ B) → (C = A ∨ C = B)))
16	1, 15	syl6ib 185	1 ⊢ ((A ∈ Cℋ ∧ B ∈ Cℋ ∧ C ∈ Cℋ ) → (A ⋖ B → ((A ⊆ C ∧ C ⊆ B) → (C = A ∨ C = B))))

Syntax hints:  ¬ wn 1   → wi 2   ∨ wo 195   ∧ wa 196   ∧ w3a 581   = wceq 1091   ∈ wcel 1092   ⊆ wss 1487   ⊂ wpss 1488   class class class wbr 2054   C_ℋ cch 4968   ⋖ ccv 4981

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3an 583  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ne 1192  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-pss 1494  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-opab 2098  df-cv 5712

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