Theorem cvnbtwn3 30659

Description: The covers relation implies no in-betweenness. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)

Assertion

Ref	Expression
cvnbtwn3	⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) → (𝐴 ⋖ℋ 𝐵 → ((𝐴 ⊆ 𝐶 ∧ 𝐶 ⊊ 𝐵) → 𝐶 = 𝐴)))

Proof of Theorem cvnbtwn3

Step	Hyp	Ref	Expression
1		cvnbtwn 30657	. 2 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) → (𝐴 ⋖ℋ 𝐵 → ¬ (𝐴 ⊊ 𝐶 ∧ 𝐶 ⊊ 𝐵)))
2		iman 402	. . 3 ⊢ (((𝐴 ⊆ 𝐶 ∧ 𝐶 ⊊ 𝐵) → 𝐴 = 𝐶) ↔ ¬ ((𝐴 ⊆ 𝐶 ∧ 𝐶 ⊊ 𝐵) ∧ ¬ 𝐴 = 𝐶))
3		eqcom 2747	. . . 4 ⊢ (𝐶 = 𝐴 ↔ 𝐴 = 𝐶)
4	3	imbi2i 336	. . 3 ⊢ (((𝐴 ⊆ 𝐶 ∧ 𝐶 ⊊ 𝐵) → 𝐶 = 𝐴) ↔ ((𝐴 ⊆ 𝐶 ∧ 𝐶 ⊊ 𝐵) → 𝐴 = 𝐶))
5		dfpss2 4025	. . . . . 6 ⊢ (𝐴 ⊊ 𝐶 ↔ (𝐴 ⊆ 𝐶 ∧ ¬ 𝐴 = 𝐶))
6	5	anbi1i 624	. . . . 5 ⊢ ((𝐴 ⊊ 𝐶 ∧ 𝐶 ⊊ 𝐵) ↔ ((𝐴 ⊆ 𝐶 ∧ ¬ 𝐴 = 𝐶) ∧ 𝐶 ⊊ 𝐵))
7		an32 643	. . . . 5 ⊢ (((𝐴 ⊆ 𝐶 ∧ ¬ 𝐴 = 𝐶) ∧ 𝐶 ⊊ 𝐵) ↔ ((𝐴 ⊆ 𝐶 ∧ 𝐶 ⊊ 𝐵) ∧ ¬ 𝐴 = 𝐶))
8	6, 7	bitri 274	. . . 4 ⊢ ((𝐴 ⊊ 𝐶 ∧ 𝐶 ⊊ 𝐵) ↔ ((𝐴 ⊆ 𝐶 ∧ 𝐶 ⊊ 𝐵) ∧ ¬ 𝐴 = 𝐶))
9	8	notbii 320	. . 3 ⊢ (¬ (𝐴 ⊊ 𝐶 ∧ 𝐶 ⊊ 𝐵) ↔ ¬ ((𝐴 ⊆ 𝐶 ∧ 𝐶 ⊊ 𝐵) ∧ ¬ 𝐴 = 𝐶))
10	2, 4, 9	3bitr4ri 304	. 2 ⊢ (¬ (𝐴 ⊊ 𝐶 ∧ 𝐶 ⊊ 𝐵) ↔ ((𝐴 ⊆ 𝐶 ∧ 𝐶 ⊊ 𝐵) → 𝐶 = 𝐴))
11	1, 10	syl6ib 250	1 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) → (𝐴 ⋖ℋ 𝐵 → ((𝐴 ⊆ 𝐶 ∧ 𝐶 ⊊ 𝐵) → 𝐶 = 𝐴)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   ∧ w3a 1086   = wceq 1542   ∈ wcel 2110   ⊆ wss 3892   ⊊ wpss 3893   class class class wbr 5079   C_ℋ cch 29300   ⋖_ℋ ccv 29335

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-ext 2711  ax-sep 5227  ax-nul 5234  ax-pr 5356

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2072  df-clab 2718  df-cleq 2732  df-clel 2818  df-ne 2946  df-ral 3071  df-rex 3072  df-rab 3075  df-v 3433  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-pss 3911  df-nul 4263  df-if 4466  df-sn 4568  df-pr 4570  df-op 4574  df-br 5080  df-opab 5142  df-cv 30650

This theorem is referenced by:  atcveq0  30719  atcvatlem  30756