Theorem cvntr 32388

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 32381	. . 3 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 ⋖ℋ 𝐵 → 𝐴 ⊊ 𝐵))
2	1	3adant3 1138	. 2 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) → (𝐴 ⋖ℋ 𝐵 → 𝐴 ⊊ 𝐵))
3		cvpss 32381	. . 3 ⊢ ((𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) → (𝐵 ⋖ℋ 𝐶 → 𝐵 ⊊ 𝐶))
4	3	3adant1 1136	. 2 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) → (𝐵 ⋖ℋ 𝐶 → 𝐵 ⊊ 𝐶))
5		cvnbtwn 32382	. . . 4 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 ⋖ℋ 𝐶 → ¬ (𝐴 ⊊ 𝐵 ∧ 𝐵 ⊊ 𝐶)))
6	5	3com23 1132	. . 3 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) → (𝐴 ⋖ℋ 𝐶 → ¬ (𝐴 ⊊ 𝐵 ∧ 𝐵 ⊊ 𝐶)))
7	6	con2d 134	. 2 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) → ((𝐴 ⊊ 𝐵 ∧ 𝐵 ⊊ 𝐶) → ¬ 𝐴 ⋖ℋ 𝐶))
8	2, 4, 7	syl2and 614	1 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) → ((𝐴 ⋖ℋ 𝐵 ∧ 𝐵 ⋖ℋ 𝐶) → ¬ 𝐴 ⋖ℋ 𝐶))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   ∧ w3a 1092   ∈ wcel 2119   ⊊ wpss 3891   class class class wbr 5079   C_ℋ cch 31025   ⋖_ℋ ccv 31060

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712  ax-sep 5225  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-ne 2936  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-br 5080  df-opab 5142  df-cv 32375

This theorem is referenced by:  atcv0eq  32475