Theorem cvnbtwnt 5718

Description: The covering relation implies no in-betweenness.

Assertion

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

Proof of Theorem cvnbtwnt

Step	Hyp	Ref	Expression
1		cvbrt 5714	. . . . 5 |- ((A e. CH /\ B e. CH) -> (A <o B <-> (A (. B /\ -. E.x e. CH (A (. x /\ x (. B))))
2		psseq2 1560	. . . . . . . . . . 11 |- (x = C -> (A (. x <-> A (. C))
3		psseq1 1559	. . . . . . . . . . 11 |- (x = C -> (x (. B <-> C (. B))
4	2, 3	anbi12d 476	. . . . . . . . . 10 |- (x = C -> ((A (. x /\ x (. B) <-> (A (. C /\ C (. B)))
5	4	rcla4ev 1403	. . . . . . . . 9 |- ((C e. CH /\ (A (. C /\ C (. B)) -> E.x e. CH (A (. x /\ x (. B))
6	5	exp 291	. . . . . . . 8 |- (C e. CH -> ((A (. C /\ C (. B) -> E.x e. CH (A (. x /\ x (. B)))
7	6	con3d 87	. . . . . . 7 |- (C e. CH -> (-. E.x e. CH (A (. x /\ x (. B) -> -. (A (. C /\ C (. B)))
8	7	com12 13	. . . . . 6 |- (-. E.x e. CH (A (. x /\ x (. B) -> (C e. CH -> -. (A (. C /\ C (. B)))
9	8	adantl 305	. . . . 5 |- ((A (. B /\ -. E.x e. CH (A (. x /\ x (. B)) -> (C e. CH -> -. (A (. C /\ C (. B)))
10	1, 9	syl6bi 187	. . . 4 |- ((A e. CH /\ B e. CH) -> (A <o B -> (C e. CH -> -. (A (. C /\ C (. B))))
11	10	com23 32	. . 3 |- ((A e. CH /\ B e. CH) -> (C e. CH -> (A <o B -> -. (A (. C /\ C (. B))))
12	11	imp 277	. 2 |- (((A e. CH /\ B e. CH) /\ C e. CH) -> (A <o B -> -. (A (. C /\ C (. B)))
13	12	3impa 609	1 |- ((A e. CH /\ B e. CH /\ C e. CH) -> (A <o B -> -. (A (. C /\ C (. B)))

Syntax hints:  -. wn 1   -> wi 2   /\ wa 196   /\ w3a 581   = wceq 1091   e. wcel 1092  E.wrex 1202   (. wpss 1488   class class class wbr 2054  CHcch 4968   <o ccv 4981

This theorem is referenced by:  cvnbtwn2t 5719  cvnbtwn3t 5720  cvnbtwn4t 5721  cvntrt 5724

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3an 583  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ne 1192  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-pss 1494  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-opab 2098  df-cv 5712

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