Theorem cvnbtwn 23742

Description: The covers relation implies no in-betweenness. (Contributed by NM, 12-Jun-2004.) (New usage is discouraged.)

Assertion

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

Proof of Theorem cvnbtwn

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cvbr 23738	. . . 4 |- ( ( A e. CH /\ B e. CH ) -> ( A <oH B <-> ( A C. B /\ -. E. x e. CH ( A C. x /\ x C. B ) ) ) )
2		psseq2 3395	. . . . . . . . 9 |- ( x = C -> ( A C. x <-> A C. C ) )
3		psseq1 3394	. . . . . . . . 9 |- ( x = C -> ( x C. B <-> C C. B ) )
4	2, 3	anbi12d 692	. . . . . . . 8 |- ( x = C -> ( ( A C. x /\ x C. B ) <-> ( A C. C /\ C C. B ) ) )
5	4	rspcev 3012	. . . . . . 7 |- ( ( C e. CH /\ ( A C. C /\ C C. B ) ) -> E. x e. CH ( A C. x /\ x C. B ) )
6	5	ex 424	. . . . . 6 |- ( C e. CH -> ( ( A C. C /\ C C. B ) -> E. x e. CH ( A C. x /\ x C. B ) ) )
7	6	con3rr3 130	. . . . 5 |- ( -. E. x e. CH ( A C. x /\ x C. B ) -> ( C e. CH -> -. ( A C. C /\ C C. B ) ) )
8	7	adantl 453	. . . 4 |- ( ( A C. B /\ -. E. x e. CH ( A C. x /\ x C. B ) ) -> ( C e. CH -> -. ( A C. C /\ C C. B ) ) )
9	1, 8	syl6bi 220	. . 3 |- ( ( A e. CH /\ B e. CH ) -> ( A <oH B -> ( C e. CH -> -. ( A C. C /\ C C. B ) ) ) )
10	9	com23 74	. 2 |- ( ( A e. CH /\ B e. CH ) -> ( C e. CH -> ( A <oH B -> -. ( A C. C /\ C C. B ) ) ) )
11	10	3impia 1150	1 |- ( ( A e. CH /\ B e. CH /\ C e. CH ) -> ( A <oH B -> -. ( A C. C /\ C C. B ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 359   /\ w3a 936   = wceq 1649   e. wcel 1721   E.wrex 2667   C. wpss 3281   class class class wbr 4172   CHcch 22385   <oH ccv 22420

This theorem is referenced by:  cvnbtwn2  23743  cvnbtwn3  23744  cvnbtwn4  23745  cvntr  23748

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-br 4173  df-opab 4227  df-cv 23735