Theorem cvnbtwn3t 10210

Description: The covering relation implies no in-betweenness.

Assertion

Ref	Expression
cvnbtwn3t	|- ((A e. CH /\ B e. CH /\ C e. CH) -> (A <o B -> ((A (_ C /\ C (. B) -> C = A)))

Proof of Theorem cvnbtwn3t

Step	Hyp	Ref	Expression
1		cvnbtwnt 10208	. 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) -> A = C) <-> -. ((A (_ C /\ C (. B) /\ -. A = C))
3		eqcom 1480	. . . 4 |- (C = A <-> A = C)
4	3	imbi2i 185	. . 3 |- (((A (_ C /\ C (. B) -> C = A) <-> ((A (_ C /\ C (. B) -> A = C))
5		dfpss2 2136	. . . . . 6 |- (A (. C <-> (A (_ C /\ -. A = C))
6	5	anbi1i 483	. . . . 5 |- ((A (. C /\ C (. B) <-> ((A (_ C /\ -. A = C) /\ C (. B))
7		an23 487	. . . . 5 |- (((A (_ C /\ -. A = C) /\ C (. B) <-> ((A (_ C /\ C (. B) /\ -. A = C))
8	6, 7	bitr 173	. . . 4 |- ((A (. C /\ C (. B) <-> ((A (_ C /\ C (. B) /\ -. A = C))
9	8	negbii 187	. . 3 |- (-. (A (. C /\ C (. B) <-> -. ((A (_ C /\ C (. B) /\ -. A = C))
10	2, 4, 9	3bitr4r 184	. 2 |- (-. (A (. C /\ C (. B) <-> ((A (_ C /\ C (. B) -> C = A))
11	1, 10	syl6ib 212	1 |- ((A e. CH /\ B e. CH /\ C e. CH) -> (A <o B -> ((A (_ C /\ C (. B) -> C = A)))

Syntax hints:  -. wn 2   -> wi 3   /\ wa 223   /\ w3a 777   = wceq 958   e. wcel 960   (_ wss 2050   (. wpss 2051   class class class wbr 2624  CHcch 8793   <o ccv 8829

This theorem is referenced by:  atcveq0 10270  atcvatlem 10307

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-rex 1653  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-pss 2058  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-br 2625  df-opab 2672  df-cv 10201

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