Theorem cvnbtwn4 32903

Description: The covers relation implies no in-betweenness. Part of proof of Lemma 7.5.1 of [MaedaMaeda] p. 31. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)

Assertion

Ref	Expression
cvnbtwn4	⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) → (𝐴 ⋖ℋ 𝐵 → ((𝐴 ⊆ 𝐶 ∧ 𝐶 ⊆ 𝐵) → (𝐶 = 𝐴 ∨ 𝐶 = 𝐵))))

Proof of Theorem cvnbtwn4

Step	Hyp	Ref	Expression
1		cvnbtwn 32900	. 2 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) → (𝐴 ⋖ℋ 𝐵 → ¬ (𝐴 ⊊ 𝐶 ∧ 𝐶 ⊊ 𝐵)))
2		iman 407	. . 3 ⊢ (((𝐴 ⊆ 𝐶 ∧ 𝐶 ⊆ 𝐵) → (𝐶 = 𝐴 ∨ 𝐶 = 𝐵)) ↔ ¬ ((𝐴 ⊆ 𝐶 ∧ 𝐶 ⊆ 𝐵) ∧ ¬ (𝐶 = 𝐴 ∨ 𝐶 = 𝐵)))
3		an4 669	. . . . 5 ⊢ (((𝐴 ⊆ 𝐶 ∧ 𝐶 ⊆ 𝐵) ∧ (¬ 𝐴 = 𝐶 ∧ ¬ 𝐶 = 𝐵)) ↔ ((𝐴 ⊆ 𝐶 ∧ ¬ 𝐴 = 𝐶) ∧ (𝐶 ⊆ 𝐵 ∧ ¬ 𝐶 = 𝐵)))
4		ioran 999	. . . . . . 7 ⊢ (¬ (𝐶 = 𝐴 ∨ 𝐶 = 𝐵) ↔ (¬ 𝐶 = 𝐴 ∧ ¬ 𝐶 = 𝐵))
5		eqcom 2770	. . . . . . . . 9 ⊢ (𝐶 = 𝐴 ↔ 𝐴 = 𝐶)
6	5	notbii 323	. . . . . . . 8 ⊢ (¬ 𝐶 = 𝐴 ↔ ¬ 𝐴 = 𝐶)
7	6	anbi1i 636	. . . . . . 7 ⊢ ((¬ 𝐶 = 𝐴 ∧ ¬ 𝐶 = 𝐵) ↔ (¬ 𝐴 = 𝐶 ∧ ¬ 𝐶 = 𝐵))
8	4, 7	bitri 278	. . . . . 6 ⊢ (¬ (𝐶 = 𝐴 ∨ 𝐶 = 𝐵) ↔ (¬ 𝐴 = 𝐶 ∧ ¬ 𝐶 = 𝐵))
9	8	anbi2i 635	. . . . 5 ⊢ (((𝐴 ⊆ 𝐶 ∧ 𝐶 ⊆ 𝐵) ∧ ¬ (𝐶 = 𝐴 ∨ 𝐶 = 𝐵)) ↔ ((𝐴 ⊆ 𝐶 ∧ 𝐶 ⊆ 𝐵) ∧ (¬ 𝐴 = 𝐶 ∧ ¬ 𝐶 = 𝐵)))
10		dfpss2 4017	. . . . . 6 ⊢ (𝐴 ⊊ 𝐶 ↔ (𝐴 ⊆ 𝐶 ∧ ¬ 𝐴 = 𝐶))
11		dfpss2 4017	. . . . . 6 ⊢ (𝐶 ⊊ 𝐵 ↔ (𝐶 ⊆ 𝐵 ∧ ¬ 𝐶 = 𝐵))
12	10, 11	anbi12i 640	. . . . 5 ⊢ ((𝐴 ⊊ 𝐶 ∧ 𝐶 ⊊ 𝐵) ↔ ((𝐴 ⊆ 𝐶 ∧ ¬ 𝐴 = 𝐶) ∧ (𝐶 ⊆ 𝐵 ∧ ¬ 𝐶 = 𝐵)))
13	3, 9, 12	3bitr4i 306	. . . 4 ⊢ (((𝐴 ⊆ 𝐶 ∧ 𝐶 ⊆ 𝐵) ∧ ¬ (𝐶 = 𝐴 ∨ 𝐶 = 𝐵)) ↔ (𝐴 ⊊ 𝐶 ∧ 𝐶 ⊊ 𝐵))
14	13	notbii 323	. . 3 ⊢ (¬ ((𝐴 ⊆ 𝐶 ∧ 𝐶 ⊆ 𝐵) ∧ ¬ (𝐶 = 𝐴 ∨ 𝐶 = 𝐵)) ↔ ¬ (𝐴 ⊊ 𝐶 ∧ 𝐶 ⊊ 𝐵))
15	2, 14	bitr2i 279	. 2 ⊢ (¬ (𝐴 ⊊ 𝐶 ∧ 𝐶 ⊊ 𝐵) ↔ ((𝐴 ⊆ 𝐶 ∧ 𝐶 ⊆ 𝐵) → (𝐶 = 𝐴 ∨ 𝐶 = 𝐵)))
16	1, 15	imbitrdi 254	1 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) → (𝐴 ⋖ℋ 𝐵 → ((𝐴 ⊆ 𝐶 ∧ 𝐶 ⊆ 𝐵) → (𝐶 = 𝐴 ∨ 𝐶 = 𝐵))))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1569   ∈ wcel 2146   ⊆ wss 3899   ⊊ wpss 4015   class class class wbr 5105   C_ℋ cch 31543   ⋖_ℋ ccv 31578

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 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-ext 2735  ax-sep 5251  ax-pr 5394

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-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3416  df-v 3455  df-ss 3900  df-pss 4016  df-dif 4049  df-in 4093  df-un 4153  df-nul 4281  df-if 4485  df-sn 4588  df-pr 4590  df-op 4838  df-br 5106  df-opab 5170  df-cv 32893

This theorem is used by:  cvmdi  32938