Theorem cvnbtwn2t 10205

Description: The covering relation implies no in-betweenness.

Assertion

Ref	Expression
cvnbtwn2t	|- ((A e. CH /\ B e. CH /\ C e. CH) -> (A <o B -> ((A (. C /\ C (_ B) -> C = B)))

Proof of Theorem cvnbtwn2t

Step	Hyp	Ref	Expression
1		cvnbtwnt 10204	. 2 |- ((A e. CH /\ B e. CH /\ C e. CH) -> (A <o B -> -. (A (. C /\ C (. B)))
2		iman 237	. . 3 |- (((A (. C /\ C (_ B) -> C = B) <-> -. ((A (. C /\ C (_ B) /\ -. C = B))
3		anass 439	. . . . 5 |- (((A (. C /\ C (_ B) /\ -. C = B) <-> (A (. C /\ (C (_ B /\ -. C = B)))
4		dfpss2 2131	. . . . . 6 |- (C (. B <-> (C (_ B /\ -. C = B))
5	4	anbi2i 480	. . . . 5 |- ((A (. C /\ C (. B) <-> (A (. C /\ (C (_ B /\ -. C = B)))
6	3, 5	bitr4 176	. . . 4 |- (((A (. C /\ C (_ B) /\ -. C = B) <-> (A (. C /\ C (. B))
7	6	negbii 187	. . 3 |- (-. ((A (. C /\ C (_ B) /\ -. C = B) <-> -. (A (. C /\ C (. B))
8	2, 7	bitr2 174	. 2 |- (-. (A (. C /\ C (. B) <-> ((A (. C /\ C (_ B) -> C = B))
9	1, 8	syl6ib 212	1 |- ((A e. CH /\ B e. CH /\ C e. CH) -> (A <o B -> ((A (. C /\ C (_ B) -> C = B)))

Syntax hints:  -. wn 2   -> wi 3   /\ wa 223   /\ w3a 774   = wceq 955   e. wcel 957   (_ wss 2045   (. wpss 2046   class class class wbr 2616  CHcch 8782   <o ccv 8818

This theorem is referenced by:  cvat 10284  cvexchlem 10286  atexcht 10299

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

Copyright terms: Public domain