Theorem cvntrt 5724

Description: The covering relation is not transitive.

Assertion

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

Proof of Theorem cvntrt

Step	Hyp	Ref	Expression
1		cvpsst 5717	. . . 4 ⊢ ((A ∈ Cℋ ∧ B ∈ Cℋ ) → (A ⋖ B → A ⊂ B))
2	1	3adant3 599	. . 3 ⊢ ((A ∈ Cℋ ∧ B ∈ Cℋ ∧ C ∈ Cℋ ) → (A ⋖ B → A ⊂ B))
3		cvpsst 5717	. . . 4 ⊢ ((B ∈ Cℋ ∧ C ∈ Cℋ ) → (B ⋖ C → B ⊂ C))
4	3	3adant1 597	. . 3 ⊢ ((A ∈ Cℋ ∧ B ∈ Cℋ ∧ C ∈ Cℋ ) → (B ⋖ C → B ⊂ C))
5	2, 4	anim12d 431	. 2 ⊢ ((A ∈ Cℋ ∧ B ∈ Cℋ ∧ C ∈ Cℋ ) → ((A ⋖ B ∧ B ⋖ C) → (A ⊂ B ∧ B ⊂ C)))
6		cvnbtwnt 5718	. . . 4 ⊢ ((A ∈ Cℋ ∧ C ∈ Cℋ ∧ B ∈ Cℋ ) → (A ⋖ C → ¬ (A ⊂ B ∧ B ⊂ C)))
7	6	3com23 616	. . 3 ⊢ ((A ∈ Cℋ ∧ B ∈ Cℋ ∧ C ∈ Cℋ ) → (A ⋖ C → ¬ (A ⊂ B ∧ B ⊂ C)))
8	7	con2d 83	. 2 ⊢ ((A ∈ Cℋ ∧ B ∈ Cℋ ∧ C ∈ Cℋ ) → ((A ⊂ B ∧ B ⊂ C) → ¬ A ⋖ C))
9	5, 8	syld 27	1 ⊢ ((A ∈ Cℋ ∧ B ∈ Cℋ ∧ C ∈ Cℋ ) → ((A ⋖ B ∧ B ⋖ C) → ¬ A ⋖ C))

Syntax hints:  ¬ wn 1   → wi 2   ∧ wa 196   ∧ w3a 581   ∈ wcel 1092   ⊂ wpss 1488   class class class wbr 2054   C_ℋ cch 4968   ⋖ ccv 4981

This theorem is referenced by:  atcv0eq 5767

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3an 583  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ne 1192  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-pss 1494  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-opab 2098  df-cv 5712

metamath.org