Theorem cvnbtwn 32945

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

Assertion

Ref	Expression
cvnbtwn	⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) → (𝐴 ⋖ℋ 𝐵 → ¬ (𝐴 ⊊ 𝐶 ∧ 𝐶 ⊊ 𝐵)))

Proof of Theorem cvnbtwn

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cvbr 32941	. . . 4 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 ⋖ℋ 𝐵 ↔ (𝐴 ⊊ 𝐵 ∧ ¬ ∃𝑥 ∈ Cℋ (𝐴 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝐵))))
2		psseq2 4021	. . . . . . . . 9 ⊢ (𝑥 = 𝐶 → (𝐴 ⊊ 𝑥 ↔ 𝐴 ⊊ 𝐶))
3		psseq1 4020	. . . . . . . . 9 ⊢ (𝑥 = 𝐶 → (𝑥 ⊊ 𝐵 ↔ 𝐶 ⊊ 𝐵))
4	2, 3	anbi12d 644	. . . . . . . 8 ⊢ (𝑥 = 𝐶 → ((𝐴 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝐵) ↔ (𝐴 ⊊ 𝐶 ∧ 𝐶 ⊊ 𝐵)))
5	4	rspcev 3580	. . . . . . 7 ⊢ ((𝐶 ∈ Cℋ ∧ (𝐴 ⊊ 𝐶 ∧ 𝐶 ⊊ 𝐵)) → ∃𝑥 ∈ Cℋ (𝐴 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝐵))
6	5	ex 418	. . . . . 6 ⊢ (𝐶 ∈ Cℋ → ((𝐴 ⊊ 𝐶 ∧ 𝐶 ⊊ 𝐵) → ∃𝑥 ∈ Cℋ (𝐴 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝐵)))
7	6	con3rr3 156	. . . . 5 ⊢ (¬ ∃𝑥 ∈ Cℋ (𝐴 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝐵) → (𝐶 ∈ Cℋ → ¬ (𝐴 ⊊ 𝐶 ∧ 𝐶 ⊊ 𝐵)))
8	7	adantl 487	. . . 4 ⊢ ((𝐴 ⊊ 𝐵 ∧ ¬ ∃𝑥 ∈ Cℋ (𝐴 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝐵)) → (𝐶 ∈ Cℋ → ¬ (𝐴 ⊊ 𝐶 ∧ 𝐶 ⊊ 𝐵)))
9	1, 8	biimtrdi 256	. . 3 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 ⋖ℋ 𝐵 → (𝐶 ∈ Cℋ → ¬ (𝐴 ⊊ 𝐶 ∧ 𝐶 ⊊ 𝐵))))
10	9	com23 87	. 2 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐶 ∈ Cℋ → (𝐴 ⋖ℋ 𝐵 → ¬ (𝐴 ⊊ 𝐶 ∧ 𝐶 ⊊ 𝐵))))
11	10	3impia 1135	1 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) → (𝐴 ⋖ℋ 𝐵 → ¬ (𝐴 ⊊ 𝐶 ∧ 𝐶 ⊊ 𝐵)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  ∃wrex 3090   ⊊ 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:  cvnbtwn2  32946  cvnbtwn3  32947  cvnbtwn4  32948  cvntr  32951