Theorem cvnbtwn4 30060

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 30057	. 2 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) → (𝐴 ⋖ℋ 𝐵 → ¬ (𝐴 ⊊ 𝐶 ∧ 𝐶 ⊊ 𝐵)))
2		iman 404	. . 3 ⊢ (((𝐴 ⊆ 𝐶 ∧ 𝐶 ⊆ 𝐵) → (𝐶 = 𝐴 ∨ 𝐶 = 𝐵)) ↔ ¬ ((𝐴 ⊆ 𝐶 ∧ 𝐶 ⊆ 𝐵) ∧ ¬ (𝐶 = 𝐴 ∨ 𝐶 = 𝐵)))
3		an4 654	. . . . 5 ⊢ (((𝐴 ⊆ 𝐶 ∧ 𝐶 ⊆ 𝐵) ∧ (¬ 𝐴 = 𝐶 ∧ ¬ 𝐶 = 𝐵)) ↔ ((𝐴 ⊆ 𝐶 ∧ ¬ 𝐴 = 𝐶) ∧ (𝐶 ⊆ 𝐵 ∧ ¬ 𝐶 = 𝐵)))
4		ioran 980	. . . . . . 7 ⊢ (¬ (𝐶 = 𝐴 ∨ 𝐶 = 𝐵) ↔ (¬ 𝐶 = 𝐴 ∧ ¬ 𝐶 = 𝐵))
5		eqcom 2828	. . . . . . . . 9 ⊢ (𝐶 = 𝐴 ↔ 𝐴 = 𝐶)
6	5	notbii 322	. . . . . . . 8 ⊢ (¬ 𝐶 = 𝐴 ↔ ¬ 𝐴 = 𝐶)
7	6	anbi1i 625	. . . . . . 7 ⊢ ((¬ 𝐶 = 𝐴 ∧ ¬ 𝐶 = 𝐵) ↔ (¬ 𝐴 = 𝐶 ∧ ¬ 𝐶 = 𝐵))
8	4, 7	bitri 277	. . . . . 6 ⊢ (¬ (𝐶 = 𝐴 ∨ 𝐶 = 𝐵) ↔ (¬ 𝐴 = 𝐶 ∧ ¬ 𝐶 = 𝐵))
9	8	anbi2i 624	. . . . 5 ⊢ (((𝐴 ⊆ 𝐶 ∧ 𝐶 ⊆ 𝐵) ∧ ¬ (𝐶 = 𝐴 ∨ 𝐶 = 𝐵)) ↔ ((𝐴 ⊆ 𝐶 ∧ 𝐶 ⊆ 𝐵) ∧ (¬ 𝐴 = 𝐶 ∧ ¬ 𝐶 = 𝐵)))
10		dfpss2 4061	. . . . . 6 ⊢ (𝐴 ⊊ 𝐶 ↔ (𝐴 ⊆ 𝐶 ∧ ¬ 𝐴 = 𝐶))
11		dfpss2 4061	. . . . . 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 305	. . . 4 ⊢ (((𝐴 ⊆ 𝐶 ∧ 𝐶 ⊆ 𝐵) ∧ ¬ (𝐶 = 𝐴 ∨ 𝐶 = 𝐵)) ↔ (𝐴 ⊊ 𝐶 ∧ 𝐶 ⊊ 𝐵))
14	13	notbii 322	. . 3 ⊢ (¬ ((𝐴 ⊆ 𝐶 ∧ 𝐶 ⊆ 𝐵) ∧ ¬ (𝐶 = 𝐴 ∨ 𝐶 = 𝐵)) ↔ ¬ (𝐴 ⊊ 𝐶 ∧ 𝐶 ⊊ 𝐵))
15	2, 14	bitr2i 278	. 2 ⊢ (¬ (𝐴 ⊊ 𝐶 ∧ 𝐶 ⊊ 𝐵) ↔ ((𝐴 ⊆ 𝐶 ∧ 𝐶 ⊆ 𝐵) → (𝐶 = 𝐴 ∨ 𝐶 = 𝐵)))
16	1, 15	syl6ib 253	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 398   ∨ wo 843   ∧ w3a 1083   = wceq 1533   ∈ wcel 2110   ⊆ wss 3935   ⊊ wpss 3936   class class class wbr 5058   C_ℋ cch 28700   ⋖_ℋ ccv 28735

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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pr 5321

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-br 5059  df-opab 5121  df-cv 30050

This theorem is referenced by:  cvmdi  30095