Theorem cvnbtwn3 32947

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 32945	. 2 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) → (𝐴 ⋖ℋ 𝐵 → ¬ (𝐴 ⊊ 𝐶 ∧ 𝐶 ⊊ 𝐵)))
2		iman 407	. . 3 ⊢ (((𝐴 ⊆ 𝐶 ∧ 𝐶 ⊊ 𝐵) → 𝐴 = 𝐶) ↔ ¬ ((𝐴 ⊆ 𝐶 ∧ 𝐶 ⊊ 𝐵) ∧ ¬ 𝐴 = 𝐶))
3		eqcom 2770	. . . 4 ⊢ (𝐶 = 𝐴 ↔ 𝐴 = 𝐶)
4	3	imbi2i 339	. . 3 ⊢ (((𝐴 ⊆ 𝐶 ∧ 𝐶 ⊊ 𝐵) → 𝐶 = 𝐴) ↔ ((𝐴 ⊆ 𝐶 ∧ 𝐶 ⊊ 𝐵) → 𝐴 = 𝐶))
5		dfpss2 4018	. . . . . 6 ⊢ (𝐴 ⊊ 𝐶 ↔ (𝐴 ⊆ 𝐶 ∧ ¬ 𝐴 = 𝐶))
6	5	anbi1i 636	. . . . 5 ⊢ ((𝐴 ⊊ 𝐶 ∧ 𝐶 ⊊ 𝐵) ↔ ((𝐴 ⊆ 𝐶 ∧ ¬ 𝐴 = 𝐶) ∧ 𝐶 ⊊ 𝐵))
7		an32 659	. . . . 5 ⊢ (((𝐴 ⊆ 𝐶 ∧ ¬ 𝐴 = 𝐶) ∧ 𝐶 ⊊ 𝐵) ↔ ((𝐴 ⊆ 𝐶 ∧ 𝐶 ⊊ 𝐵) ∧ ¬ 𝐴 = 𝐶))
8	6, 7	bitri 278	. . . 4 ⊢ ((𝐴 ⊊ 𝐶 ∧ 𝐶 ⊊ 𝐵) ↔ ((𝐴 ⊆ 𝐶 ∧ 𝐶 ⊊ 𝐵) ∧ ¬ 𝐴 = 𝐶))
9	8	notbii 323	. . 3 ⊢ (¬ (𝐴 ⊊ 𝐶 ∧ 𝐶 ⊊ 𝐵) ↔ ¬ ((𝐴 ⊆ 𝐶 ∧ 𝐶 ⊊ 𝐵) ∧ ¬ 𝐴 = 𝐶))
10	2, 4, 9	3bitr4ri 307	. 2 ⊢ (¬ (𝐴 ⊊ 𝐶 ∧ 𝐶 ⊊ 𝐵) ↔ ((𝐴 ⊆ 𝐶 ∧ 𝐶 ⊊ 𝐵) → 𝐶 = 𝐴))
11	1, 10	imbitrdi 254	1 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) → (𝐴 ⋖ℋ 𝐵 → ((𝐴 ⊆ 𝐶 ∧ 𝐶 ⊊ 𝐵) → 𝐶 = 𝐴)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147   ⊆ wss 3900   ⊊ wpss 4016   class class class wbr 5108   C_ℋ cch 31588   ⋖_ℋ ccv 31623

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5255  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-br 5109  df-opab 5174  df-cv 32938

This theorem is used by:  atcveq0  33007  atcvatlem  33044