Theorem cvnbtwn 32711

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 32707	. . . 4 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 ⋖ℋ 𝐵 ↔ (𝐴 ⊊ 𝐵 ∧ ¬ ∃𝑥 ∈ Cℋ (𝐴 ⊊ 𝑥 ∧ 𝑥 ⊊ 𝐵))))
2		psseq2 4046	. . . . . . . . 9 ⊢ (𝑥 = 𝐶 → (𝐴 ⊊ 𝑥 ↔ 𝐴 ⊊ 𝐶))
3		psseq1 4045	. . . . . . . . 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 3583	. . . . . . 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 2146  ∃wrex 3091   ⊊ wpss 3907   class class class wbr 5111   C_ℋ cch 31354   ⋖_ℋ ccv 31389

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 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737  ax-sep 5259  ax-pr 5406

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 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-br 5112  df-opab 5176  df-cv 32704

This theorem is used by:  cvnbtwn2  32712  cvnbtwn3  32713  cvnbtwn4  32714  cvntr  32717