Theorem cvnsym 29149

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

Assertion

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

Proof of Theorem cvnsym

Step	Hyp	Ref	Expression
1		cvpss 29144	. 2 |- ( ( A e. CH /\ B e. CH ) -> ( A <oH B -> A C. B ) )
2		cvpss 29144	. . . . 5 |- ( ( B e. CH /\ A e. CH ) -> ( B <oH A -> B C. A ) )
3	2	ancoms 469	. . . 4 |- ( ( A e. CH /\ B e. CH ) -> ( B <oH A -> B C. A ) )
4		pssn2lp 3708	. . . . 5 |- -. ( B C. A /\ A C. B )
5	4	imnani 439	. . . 4 |- ( B C. A -> -. A C. B )
6	3, 5	syl6 35	. . 3 |- ( ( A e. CH /\ B e. CH ) -> ( B <oH A -> -. A C. B ) )
7	6	con2d 129	. 2 |- ( ( A e. CH /\ B e. CH ) -> ( A C. B -> -. B <oH A ) )
8	1, 7	syld 47	1 |- ( ( A e. CH /\ B e. CH ) -> ( A <oH B -> -. B <oH A ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   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:  cvnref  29150