Theorem cvnbtwn4t 10216

Description: The covering relation implies no in-betweenness. Part of proof of Lemma 7.5.1 of [MaedaMaeda] p. 31.

Assertion

Ref	Expression
cvnbtwn4t	|- ((A e. CH /\ B e. CH /\ C e. CH) -> (A <o B -> ((A (_ C /\ C (_ B) -> (C = A \/ C = B))))

Proof of Theorem cvnbtwn4t

Step	Hyp	Ref	Expression
1		cvnbtwnt 10213	. 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 = A \/ C = B)) <-> -. ((A (_ C /\ C (_ B) /\ -. (C = A \/ C = B)))
3		an4 506	. . . . 5 |- (((A (_ C /\ C (_ B) /\ (-. A = C /\ -. C = B)) <-> ((A (_ C /\ -. A = C) /\ (C (_ B /\ -. C = B)))
4		ioran 306	. . . . . . 7 |- (-. (C = A \/ C = B) <-> (-. C = A /\ -. C = B))
5		eqcom 1477	. . . . . . . . 9 |- (C = A <-> A = C)
6	5	negbii 187	. . . . . . . 8 |- (-. C = A <-> -. A = C)
7	6	anbi1i 481	. . . . . . 7 |- ((-. C = A /\ -. C = B) <-> (-. A = C /\ -. C = B))
8	4, 7	bitr 173	. . . . . 6 |- (-. (C = A \/ C = B) <-> (-. A = C /\ -. C = B))
9	8	anbi2i 480	. . . . 5 |- (((A (_ C /\ C (_ B) /\ -. (C = A \/ C = B)) <-> ((A (_ C /\ C (_ B) /\ (-. A = C /\ -. C = B)))
10		dfpss2 2133	. . . . . 6 |- (A (. C <-> (A (_ C /\ -. A = C))
11		dfpss2 2133	. . . . . 6 |- (C (. B <-> (C (_ B /\ -. C = B))
12	10, 11	anbi12i 482	. . . . 5 |- ((A (. C /\ C (. B) <-> ((A (_ C /\ -. A = C) /\ (C (_ B /\ -. C = B)))
13	3, 9, 12	3bitr4 183	. . . 4 |- (((A (_ C /\ C (_ B) /\ -. (C = A \/ C = B)) <-> (A (. C /\ C (. B))
14	13	negbii 187	. . 3 |- (-. ((A (_ C /\ C (_ B) /\ -. (C = A \/ C = B)) <-> -. (A (. C /\ C (. B))
15	2, 14	bitr2 174	. 2 |- (-. (A (. C /\ C (. B) <-> ((A (_ C /\ C (_ B) -> (C = A \/ C = B)))
16	1, 15	syl6ib 212	1 |- ((A e. CH /\ B e. CH /\ C e. CH) -> (A <o B -> ((A (_ C /\ C (_ B) -> (C = A \/ C = B))))

Syntax hints:  -. wn 2   -> wi 3   \/ wo 222   /\ wa 223   /\ w3a 775   = wceq 956   e. wcel 958   (_ wss 2047   (. wpss 2048   class class class wbr 2619  CHcch 8798   <o ccv 8834

This theorem is referenced by:  cvmd 10251

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-pss 2055  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-br 2620  df-opab 2667  df-cv 10206

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