Theorem cvnbtwn3 32320

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 32318	. 2 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) → (𝐴 ⋖ℋ 𝐵 → ¬ (𝐴 ⊊ 𝐶 ∧ 𝐶 ⊊ 𝐵)))
2		iman 401	. . 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 4111	. . . . . 6 ⊢ (𝐴 ⊊ 𝐶 ↔ (𝐴 ⊆ 𝐶 ∧ ¬ 𝐴 = 𝐶))
6	5	anbi1i 623	. . . . 5 ⊢ ((𝐴 ⊊ 𝐶 ∧ 𝐶 ⊊ 𝐵) ↔ ((𝐴 ⊆ 𝐶 ∧ ¬ 𝐴 = 𝐶) ∧ 𝐶 ⊊ 𝐵))
7		an32 645	. . . . 5 ⊢ (((𝐴 ⊆ 𝐶 ∧ ¬ 𝐴 = 𝐶) ∧ 𝐶 ⊊ 𝐵) ↔ ((𝐴 ⊆ 𝐶 ∧ 𝐶 ⊊ 𝐵) ∧ ¬ 𝐴 = 𝐶))
8	6, 7	bitri 275	. . . 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	imbitrdi 251	1 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) → (𝐴 ⋖ℋ 𝐵 → ((𝐴 ⊆ 𝐶 ∧ 𝐶 ⊊ 𝐵) → 𝐶 = 𝐴)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108   ⊆ wss 3976   ⊊ wpss 3977   class class class wbr 5166   C_ℋ cch 30961   ⋖_ℋ ccv 30996

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-cv 32311

This theorem is referenced by:  atcveq0  32380  atcvatlem  32417