Theorem cvntr 29151

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

Assertion

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

Proof of Theorem cvntr

Step	Hyp	Ref	Expression
1		cvpss 29144	. . 3 |- ( ( A e. CH /\ B e. CH ) -> ( A <oH B -> A C. B ) )
2	1	3adant3 1081	. 2 |- ( ( A e. CH /\ B e. CH /\ C e. CH ) -> ( A <oH B -> A C. B ) )
3		cvpss 29144	. . 3 |- ( ( B e. CH /\ C e. CH ) -> ( B <oH C -> B C. C ) )
4	3	3adant1 1079	. 2 |- ( ( A e. CH /\ B e. CH /\ C e. CH ) -> ( B <oH C -> B C. C ) )
5		cvnbtwn 29145	. . . 4 |- ( ( A e. CH /\ C e. CH /\ B e. CH ) -> ( A <oH C -> -. ( A C. B /\ B C. C ) ) )
6	5	3com23 1271	. . 3 |- ( ( A e. CH /\ B e. CH /\ C e. CH ) -> ( A <oH C -> -. ( A C. B /\ B C. C ) ) )
7	6	con2d 129	. 2 |- ( ( A e. CH /\ B e. CH /\ C e. CH ) -> ( ( A C. B /\ B C. C ) -> -. A <oH C ) )
8	2, 4, 7	syl2and 500	1 |- ( ( A e. CH /\ B e. CH /\ C e. CH ) -> ( ( A <oH B /\ B <oH C ) -> -. A <oH C ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   /\ w3a 1037   e. wcel 1990   C. wpss 3575   class class class wbr 4653   CHcch 27786   <oH ccv 27821

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-cv 29138

This theorem is referenced by:  atcv0eq  29238