Theorem cvnbtwn4 32318

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 32315	. 2 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) → (𝐴 ⋖ℋ 𝐵 → ¬ (𝐴 ⊊ 𝐶 ∧ 𝐶 ⊊ 𝐵)))
2		iman 401	. . 3 ⊢ (((𝐴 ⊆ 𝐶 ∧ 𝐶 ⊆ 𝐵) → (𝐶 = 𝐴 ∨ 𝐶 = 𝐵)) ↔ ¬ ((𝐴 ⊆ 𝐶 ∧ 𝐶 ⊆ 𝐵) ∧ ¬ (𝐶 = 𝐴 ∨ 𝐶 = 𝐵)))
3		an4 656	. . . . 5 ⊢ (((𝐴 ⊆ 𝐶 ∧ 𝐶 ⊆ 𝐵) ∧ (¬ 𝐴 = 𝐶 ∧ ¬ 𝐶 = 𝐵)) ↔ ((𝐴 ⊆ 𝐶 ∧ ¬ 𝐴 = 𝐶) ∧ (𝐶 ⊆ 𝐵 ∧ ¬ 𝐶 = 𝐵)))
4		ioran 985	. . . . . . 7 ⊢ (¬ (𝐶 = 𝐴 ∨ 𝐶 = 𝐵) ↔ (¬ 𝐶 = 𝐴 ∧ ¬ 𝐶 = 𝐵))
5		eqcom 2742	. . . . . . . . 9 ⊢ (𝐶 = 𝐴 ↔ 𝐴 = 𝐶)
6	5	notbii 320	. . . . . . . 8 ⊢ (¬ 𝐶 = 𝐴 ↔ ¬ 𝐴 = 𝐶)
7	6	anbi1i 624	. . . . . . 7 ⊢ ((¬ 𝐶 = 𝐴 ∧ ¬ 𝐶 = 𝐵) ↔ (¬ 𝐴 = 𝐶 ∧ ¬ 𝐶 = 𝐵))
8	4, 7	bitri 275	. . . . . 6 ⊢ (¬ (𝐶 = 𝐴 ∨ 𝐶 = 𝐵) ↔ (¬ 𝐴 = 𝐶 ∧ ¬ 𝐶 = 𝐵))
9	8	anbi2i 623	. . . . 5 ⊢ (((𝐴 ⊆ 𝐶 ∧ 𝐶 ⊆ 𝐵) ∧ ¬ (𝐶 = 𝐴 ∨ 𝐶 = 𝐵)) ↔ ((𝐴 ⊆ 𝐶 ∧ 𝐶 ⊆ 𝐵) ∧ (¬ 𝐴 = 𝐶 ∧ ¬ 𝐶 = 𝐵)))
10		dfpss2 4098	. . . . . 6 ⊢ (𝐴 ⊊ 𝐶 ↔ (𝐴 ⊆ 𝐶 ∧ ¬ 𝐴 = 𝐶))
11		dfpss2 4098	. . . . . 6 ⊢ (𝐶 ⊊ 𝐵 ↔ (𝐶 ⊆ 𝐵 ∧ ¬ 𝐶 = 𝐵))
12	10, 11	anbi12i 628	. . . . 5 ⊢ ((𝐴 ⊊ 𝐶 ∧ 𝐶 ⊊ 𝐵) ↔ ((𝐴 ⊆ 𝐶 ∧ ¬ 𝐴 = 𝐶) ∧ (𝐶 ⊆ 𝐵 ∧ ¬ 𝐶 = 𝐵)))
13	3, 9, 12	3bitr4i 303	. . . 4 ⊢ (((𝐴 ⊆ 𝐶 ∧ 𝐶 ⊆ 𝐵) ∧ ¬ (𝐶 = 𝐴 ∨ 𝐶 = 𝐵)) ↔ (𝐴 ⊊ 𝐶 ∧ 𝐶 ⊊ 𝐵))
14	13	notbii 320	. . 3 ⊢ (¬ ((𝐴 ⊆ 𝐶 ∧ 𝐶 ⊆ 𝐵) ∧ ¬ (𝐶 = 𝐴 ∨ 𝐶 = 𝐵)) ↔ ¬ (𝐴 ⊊ 𝐶 ∧ 𝐶 ⊊ 𝐵))
15	2, 14	bitr2i 276	. 2 ⊢ (¬ (𝐴 ⊊ 𝐶 ∧ 𝐶 ⊊ 𝐵) ↔ ((𝐴 ⊆ 𝐶 ∧ 𝐶 ⊆ 𝐵) → (𝐶 = 𝐴 ∨ 𝐶 = 𝐵)))
16	1, 15	imbitrdi 251	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 395   ∨ wo 847   ∧ w3a 1086   = wceq 1537   ∈ wcel 2106   ⊆ wss 3963   ⊊ wpss 3964   class class class wbr 5148   C_ℋ cch 30958   ⋖_ℋ ccv 30993

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-cv 32308

This theorem is referenced by:  cvmdi  32353