Theorem cvntr 30075

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 30068	. . 3 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 ⋖ℋ 𝐵 → 𝐴 ⊊ 𝐵))
2	1	3adant3 1129	. 2 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) → (𝐴 ⋖ℋ 𝐵 → 𝐴 ⊊ 𝐵))
3		cvpss 30068	. . 3 ⊢ ((𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) → (𝐵 ⋖ℋ 𝐶 → 𝐵 ⊊ 𝐶))
4	3	3adant1 1127	. 2 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) → (𝐵 ⋖ℋ 𝐶 → 𝐵 ⊊ 𝐶))
5		cvnbtwn 30069	. . . 4 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 ⋖ℋ 𝐶 → ¬ (𝐴 ⊊ 𝐵 ∧ 𝐵 ⊊ 𝐶)))
6	5	3com23 1123	. . 3 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) → (𝐴 ⋖ℋ 𝐶 → ¬ (𝐴 ⊊ 𝐵 ∧ 𝐵 ⊊ 𝐶)))
7	6	con2d 136	. 2 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) → ((𝐴 ⊊ 𝐵 ∧ 𝐵 ⊊ 𝐶) → ¬ 𝐴 ⋖ℋ 𝐶))
8	2, 4, 7	syl2and 610	1 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) → ((𝐴 ⋖ℋ 𝐵 ∧ 𝐵 ⋖ℋ 𝐶) → ¬ 𝐴 ⋖ℋ 𝐶))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∧ w3a 1084   ∈ wcel 2111   ⊊ wpss 3882   class class class wbr 5030   C_ℋ cch 28712   ⋖_ℋ ccv 28747

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-br 5031  df-opab 5093  df-cv 30062

This theorem is referenced by:  atcv0eq  30162