Theorem cvnbtwn2 32049

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

Assertion

Ref	Expression
cvnbtwn2	⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) → (𝐴 ⋖ℋ 𝐵 → ((𝐴 ⊊ 𝐶 ∧ 𝐶 ⊆ 𝐵) → 𝐶 = 𝐵)))

Proof of Theorem cvnbtwn2

Step	Hyp	Ref	Expression
1		cvnbtwn 32048	. 2 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) → (𝐴 ⋖ℋ 𝐵 → ¬ (𝐴 ⊊ 𝐶 ∧ 𝐶 ⊊ 𝐵)))
2		iman 401	. . 3 ⊢ (((𝐴 ⊊ 𝐶 ∧ 𝐶 ⊆ 𝐵) → 𝐶 = 𝐵) ↔ ¬ ((𝐴 ⊊ 𝐶 ∧ 𝐶 ⊆ 𝐵) ∧ ¬ 𝐶 = 𝐵))
3		anass 468	. . . . 5 ⊢ (((𝐴 ⊊ 𝐶 ∧ 𝐶 ⊆ 𝐵) ∧ ¬ 𝐶 = 𝐵) ↔ (𝐴 ⊊ 𝐶 ∧ (𝐶 ⊆ 𝐵 ∧ ¬ 𝐶 = 𝐵)))
4		dfpss2 4080	. . . . . 6 ⊢ (𝐶 ⊊ 𝐵 ↔ (𝐶 ⊆ 𝐵 ∧ ¬ 𝐶 = 𝐵))
5	4	anbi2i 622	. . . . 5 ⊢ ((𝐴 ⊊ 𝐶 ∧ 𝐶 ⊊ 𝐵) ↔ (𝐴 ⊊ 𝐶 ∧ (𝐶 ⊆ 𝐵 ∧ ¬ 𝐶 = 𝐵)))
6	3, 5	bitr4i 278	. . . 4 ⊢ (((𝐴 ⊊ 𝐶 ∧ 𝐶 ⊆ 𝐵) ∧ ¬ 𝐶 = 𝐵) ↔ (𝐴 ⊊ 𝐶 ∧ 𝐶 ⊊ 𝐵))
7	6	notbii 320	. . 3 ⊢ (¬ ((𝐴 ⊊ 𝐶 ∧ 𝐶 ⊆ 𝐵) ∧ ¬ 𝐶 = 𝐵) ↔ ¬ (𝐴 ⊊ 𝐶 ∧ 𝐶 ⊊ 𝐵))
8	2, 7	bitr2i 276	. 2 ⊢ (¬ (𝐴 ⊊ 𝐶 ∧ 𝐶 ⊊ 𝐵) ↔ ((𝐴 ⊊ 𝐶 ∧ 𝐶 ⊆ 𝐵) → 𝐶 = 𝐵))
9	1, 8	imbitrdi 250	1 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) → (𝐴 ⋖ℋ 𝐵 → ((𝐴 ⊊ 𝐶 ∧ 𝐶 ⊆ 𝐵) → 𝐶 = 𝐵)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   ⊆ wss 3943   ⊊ wpss 3944   class class class wbr 5141   C_ℋ cch 30691   ⋖_ℋ ccv 30726

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-cv 32041

This theorem is referenced by:  cvati  32128  cvexchlem  32130  atexch  32143