Theorem oppne2 25634

Description: Points lying on opposite sides of a line cannot be on the line. (Contributed by Thierry Arnoux, 3-Mar-2020.)

Hypotheses

Ref	Expression
hpg.p	|- P = ( Base ` G )
hpg.d	|- .- = ( dist ` G )
hpg.i	|- I = (Itv ` G )
hpg.o	|- O = { <. a , b >. | ( ( a e. ( P \ D ) /\ b e. ( P \ D ) ) /\ E. t e. D t e. ( a I b ) ) }
opphl.l	|- L = (LineG ` G )
opphl.d	|- ( ph -> D e. ran L )
opphl.g	|- ( ph -> G e. TarskiG )
oppcom.a	|- ( ph -> A e. P )
oppcom.b	|- ( ph -> B e. P )
oppcom.o	|- ( ph -> A O B )

Assertion

Ref	Expression
oppne2	|- ( ph -> -. B e. D )

Distinct variable groups:   D, a, b   I, a, b   P, a, b   t, A   t, B   t, D   t, G   t, L   t, I   t, O   t, P   ph, t   t, .-   t, a, b

Allowed substitution hints:   ph( a, b)   A( a, b)   B( a, b)   G( a, b)   L( a, b)   .- ( a, b)   O( a, b)

Proof of Theorem oppne2

Step	Hyp	Ref	Expression
1		oppcom.o	. . . 4 |- ( ph -> A O B )
2		hpg.p	. . . . 5 |- P = ( Base ` G )
3		hpg.d	. . . . 5 |- .- = ( dist ` G )
4		hpg.i	. . . . 5 |- I = (Itv ` G )
5		hpg.o	. . . . 5 |- O = { <. a , b >. | ( ( a e. ( P \ D ) /\ b e. ( P \ D ) ) /\ E. t e. D t e. ( a I b ) ) }
6		oppcom.a	. . . . 5 |- ( ph -> A e. P )
7		oppcom.b	. . . . 5 |- ( ph -> B e. P )
8	2, 3, 4, 5, 6, 7	islnopp 25631	. . . 4 |- ( ph -> ( A O B <-> ( ( -. A e. D /\ -. B e. D ) /\ E. t e. D t e. ( A I B ) ) ) )
9	1, 8	mpbid 222	. . 3 |- ( ph -> ( ( -. A e. D /\ -. B e. D ) /\ E. t e. D t e. ( A I B ) ) )
10	9	simpld 475	. 2 |- ( ph -> ( -. A e. D /\ -. B e. D ) )
11	10	simprd 479	1 |- ( ph -> -. B e. D )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   \ cdif 3571   class class class wbr 4653   {copab 4712   ran crn 5115   ` cfv 5888  (class class class)co 6650   Basecbs 15857   distcds 15950  TarskiGcstrkg 25329  Itvcitv 25335  LineGclng 25336

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-iota 5851  df-fv 5896  df-ov 6653

This theorem is referenced by:  opphllem1  25639  opphllem2  25640  opphl  25646  lnopp2hpgb  25655  lnperpex  25695