Theorem cvnbtwnt 10208

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 10204	. . . 4 |- ((A e. CH /\ B e. CH) -> (A <o B <-> (A (. B /\ -. E.x e. CH (A (. x /\ x (. B))))
2		psseq2 2139	. . . . . . . . . 10 |- (x = C -> (A (. x <-> A (. C))
3		psseq1 2138	. . . . . . . . . 10 |- (x = C -> (x (. B <-> C (. B))
4	2, 3	anbi12d 630	. . . . . . . . 9 |- (x = C -> ((A (. x /\ x (. B) <-> (A (. C /\ C (. B)))
5	4	rcla4ev 1880	. . . . . . . 8 |- ((C e. CH /\ (A (. C /\ C (. B)) -> E.x e. CH (A (. x /\ x (. B))
6	5	ex 373	. . . . . . 7 |- (C e. CH -> ((A (. C /\ C (. B) -> E.x e. CH (A (. x /\ x (. B)))
7	6	con3d 95	. . . . . 6 |- (C e. CH -> (-. E.x e. CH (A (. x /\ x (. B) -> -. (A (. C /\ C (. B)))
8	7	com12 11	. . . . 5 |- (-. E.x e. CH (A (. x /\ x (. B) -> (C e. CH -> -. (A (. C /\ C (. B)))
9	8	adantl 390	. . . 4 |- ((A (. B /\ -. E.x e. CH (A (. x /\ x (. B)) -> (C e. CH -> -. (A (. C /\ C (. B)))
10	1, 9	syl6bi 214	. . 3 |- ((A e. CH /\ B e. CH) -> (A <o B -> (C e. CH -> -. (A (. C /\ C (. B))))
11	10	com23 32	. 2 |- ((A e. CH /\ B e. CH) -> (C e. CH -> (A <o B -> -. (A (. C /\ C (. B))))
12	11	3impia 832	1 |- ((A e. CH /\ B e. CH /\ C e. CH) -> (A <o B -> -. (A (. C /\ C (. B)))

Syntax hints:  -. wn 2   -> wi 3   /\ wa 223   /\ w3a 777   = wceq 958   e. wcel 960  E.wrex 1649   (. wpss 2051   class class class wbr 2624  CHcch 8793   <o ccv 8829

This theorem is referenced by:  cvnbtwn2t 10209  cvnbtwn3t 10210  cvnbtwn4t 10211  cvntrt 10214

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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