Theorem cvntr 32681

Description: The covers relation is not transitive. (Contributed by NM, 26-Jun-2004.) (New usage is discouraged.)

Assertion

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

Proof of Theorem cvntr

Step	Hyp	Ref	Expression
1		cvpss 32674	. . 3 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 ⋖ℋ 𝐵 → 𝐴 ⊊ 𝐵))
2	1	3adant3 1150	. 2 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) → (𝐴 ⋖ℋ 𝐵 → 𝐴 ⊊ 𝐵))
3		cvpss 32674	. . 3 ⊢ ((𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) → (𝐵 ⋖ℋ 𝐶 → 𝐵 ⊊ 𝐶))
4	3	3adant1 1148	. 2 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) → (𝐵 ⋖ℋ 𝐶 → 𝐵 ⊊ 𝐶))
5		cvnbtwn 32675	. . . 4 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 ⋖ℋ 𝐶 → ¬ (𝐴 ⊊ 𝐵 ∧ 𝐵 ⊊ 𝐶)))
6	5	3com23 1144	. . 3 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) → (𝐴 ⋖ℋ 𝐶 → ¬ (𝐴 ⊊ 𝐵 ∧ 𝐵 ⊊ 𝐶)))
7	6	con2d 135	. 2 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) → ((𝐴 ⊊ 𝐵 ∧ 𝐵 ⊊ 𝐶) → ¬ 𝐴 ⋖ℋ 𝐶))
8	2, 4, 7	syl2and 620	1 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) → ((𝐴 ⋖ℋ 𝐵 ∧ 𝐵 ⋖ℋ 𝐶) → ¬ 𝐴 ⋖ℋ 𝐶))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∧ w3a 1103   ∈ wcel 2146   ⊊ wpss 3909   class class class wbr 5114   C_ℋ cch 31318   ⋖_ℋ ccv 31353

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 2738  ax-sep 5262  ax-pr 5409

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 2745  df-cleq 2758  df-clel 2841  df-ne 2962  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3928  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5115  df-opab 5179  df-cv 32668

This theorem is used by:  atcv0eq  32768