Theorem cvntrt 10210

Description: The covering relation is not transitive.

Assertion

Ref	Expression
cvntrt	|- ((A e. CH /\ B e. CH /\ C e. CH) -> ((A <o B /\ B <o C) -> -. A <o C))

Proof of Theorem cvntrt

Step	Hyp	Ref	Expression
1		cvnbtwnt 10204	. . . 4 |- ((A e. CH /\ C e. CH /\ B e. CH) -> (A <o C -> -. (A (. B /\ B (. C)))
2	1	3com23 838	. . 3 |- ((A e. CH /\ B e. CH /\ C e. CH) -> (A <o C -> -. (A (. B /\ B (. C)))
3	2	con2d 91	. 2 |- ((A e. CH /\ B e. CH /\ C e. CH) -> ((A (. B /\ B (. C) -> -. A <o C))
4		cvpsst 10203	. . 3 |- ((A e. CH /\ B e. CH) -> (A <o B -> A (. B))
5	4	3adant3 798	. 2 |- ((A e. CH /\ B e. CH /\ C e. CH) -> (A <o B -> A (. B))
6		cvpsst 10203	. . 3 |- ((B e. CH /\ C e. CH) -> (B <o C -> B (. C))
7	6	3adant1 796	. 2 |- ((A e. CH /\ B e. CH /\ C e. CH) -> (B <o C -> B (. C))
8	3, 5, 7	syl2and 459	1 |- ((A e. CH /\ B e. CH /\ C e. CH) -> ((A <o B /\ B <o C) -> -. A <o C))

Syntax hints:  -. wn 2   -> wi 3   /\ wa 223   /\ w3a 774   e. wcel 957   (. wpss 2046   class class class wbr 2616  CHcch 8782   <o ccv 8818

This theorem is referenced by:  atcv0eq 10297

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2700  ax-pow 2739  ax-pr 2776

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 980  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1586  df-rex 1649  df-v 1810  df-dif 2047  df-un 2048  df-in 2049  df-ss 2051  df-pss 2053  df-nul 2279  df-pw 2400  df-sn 2410  df-pr 2411  df-op 2414  df-br 2617  df-opab 2664  df-cv 10197

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