Theorem cvnbtwn4 32582

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 32579	. 2 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) → (𝐴 ⋖ℋ 𝐵 → ¬ (𝐴 ⊊ 𝐶 ∧ 𝐶 ⊊ 𝐵)))
2		iman 406	. . 3 ⊢ (((𝐴 ⊆ 𝐶 ∧ 𝐶 ⊆ 𝐵) → (𝐶 = 𝐴 ∨ 𝐶 = 𝐵)) ↔ ¬ ((𝐴 ⊆ 𝐶 ∧ 𝐶 ⊆ 𝐵) ∧ ¬ (𝐶 = 𝐴 ∨ 𝐶 = 𝐵)))
3		an4 668	. . . . 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 2776	. . . . . . . . 9 ⊢ (𝐶 = 𝐴 ↔ 𝐴 = 𝐶)
6	5	notbii 323	. . . . . . . 8 ⊢ (¬ 𝐶 = 𝐴 ↔ ¬ 𝐴 = 𝐶)
7	6	anbi1i 635	. . . . . . 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 634	. . . . 5 ⊢ (((𝐴 ⊆ 𝐶 ∧ 𝐶 ⊆ 𝐵) ∧ ¬ (𝐶 = 𝐴 ∨ 𝐶 = 𝐵)) ↔ ((𝐴 ⊆ 𝐶 ∧ 𝐶 ⊆ 𝐵) ∧ (¬ 𝐴 = 𝐶 ∧ ¬ 𝐶 = 𝐵)))
10		dfpss2 4050	. . . . . 6 ⊢ (𝐴 ⊊ 𝐶 ↔ (𝐴 ⊆ 𝐶 ∧ ¬ 𝐴 = 𝐶))
11		dfpss2 4050	. . . . . 6 ⊢ (𝐶 ⊊ 𝐵 ↔ (𝐶 ⊆ 𝐵 ∧ ¬ 𝐶 = 𝐵))
12	10, 11	anbi12i 639	. . . . 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))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   ∨ wo 860   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149   ⊆ wss 3913   ⊊ wpss 3914   class class class wbr 5113   C_ℋ cch 31222   ⋖_ℋ ccv 31257

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5261  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5114  df-opab 5178  df-cv 32572

This theorem is referenced by:  cvmdi  32617