Theorem btwndiff 32134

Description: There is always a c distinct from B such that B lies between A and c. Theorem 3.14 of [Schwabhauser] p. 32. (Contributed by Scott Fenton, 24-Sep-2013.)

Assertion

Ref	Expression
btwndiff	|- ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) -> E. c e. ( EE ` N ) ( B Btwn <. A , c >. /\ B =/= c ) )

Distinct variable groups:   A, c   B, c   N, c

Proof of Theorem btwndiff

Dummy variables u v are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		axlowdim1 25839	. . 3 |- ( N e. NN -> E. u e. ( EE ` N ) E. v e. ( EE ` N ) u =/= v )
2	1	3ad2ant1 1082	. 2 |- ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) -> E. u e. ( EE ` N ) E. v e. ( EE ` N ) u =/= v )
3		simp11 1091	. . . . . 6 |- ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( u e. ( EE ` N ) /\ v e. ( EE ` N ) ) /\ u =/= v ) -> N e. NN )
4		simp12 1092	. . . . . 6 |- ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( u e. ( EE ` N ) /\ v e. ( EE ` N ) ) /\ u =/= v ) -> A e. ( EE ` N ) )
5		simp13 1093	. . . . . 6 |- ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( u e. ( EE ` N ) /\ v e. ( EE ` N ) ) /\ u =/= v ) -> B e. ( EE ` N ) )
6		simp2l 1087	. . . . . 6 |- ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( u e. ( EE ` N ) /\ v e. ( EE ` N ) ) /\ u =/= v ) -> u e. ( EE ` N ) )
7		simp2r 1088	. . . . . 6 |- ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( u e. ( EE ` N ) /\ v e. ( EE ` N ) ) /\ u =/= v ) -> v e. ( EE ` N ) )
8		axsegcon 25807	. . . . . 6 |- ( ( N e. NN /\ ( A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( u e. ( EE ` N ) /\ v e. ( EE ` N ) ) ) -> E. c e. ( EE ` N ) ( B Btwn <. A , c >. /\ <. B , c >.Cgr <. u , v >. ) )
9	3, 4, 5, 6, 7, 8	syl122anc 1335	. . . . 5 |- ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( u e. ( EE ` N ) /\ v e. ( EE ` N ) ) /\ u =/= v ) -> E. c e. ( EE ` N ) ( B Btwn <. A , c >. /\ <. B , c >.Cgr <. u , v >. ) )
10		simpl11 1136	. . . . . . . . 9 |- ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( u e. ( EE ` N ) /\ v e. ( EE ` N ) ) /\ u =/= v ) /\ c e. ( EE ` N ) ) -> N e. NN )
11		simpl13 1138	. . . . . . . . 9 |- ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( u e. ( EE ` N ) /\ v e. ( EE ` N ) ) /\ u =/= v ) /\ c e. ( EE ` N ) ) -> B e. ( EE ` N ) )
12		simpr 477	. . . . . . . . 9 |- ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( u e. ( EE ` N ) /\ v e. ( EE ` N ) ) /\ u =/= v ) /\ c e. ( EE ` N ) ) -> c e. ( EE ` N ) )
13		simpl2l 1114	. . . . . . . . 9 |- ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( u e. ( EE ` N ) /\ v e. ( EE ` N ) ) /\ u =/= v ) /\ c e. ( EE ` N ) ) -> u e. ( EE ` N ) )
14		simpl2r 1115	. . . . . . . . 9 |- ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( u e. ( EE ` N ) /\ v e. ( EE ` N ) ) /\ u =/= v ) /\ c e. ( EE ` N ) ) -> v e. ( EE ` N ) )
15		cgrdegen 32111	. . . . . . . . 9 |- ( ( N e. NN /\ ( B e. ( EE ` N ) /\ c e. ( EE ` N ) ) /\ ( u e. ( EE ` N ) /\ v e. ( EE ` N ) ) ) -> ( <. B , c >.Cgr <. u , v >. -> ( B = c <-> u = v ) ) )
16	10, 11, 12, 13, 14, 15	syl122anc 1335	. . . . . . . 8 |- ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( u e. ( EE ` N ) /\ v e. ( EE ` N ) ) /\ u =/= v ) /\ c e. ( EE ` N ) ) -> ( <. B , c >.Cgr <. u , v >. -> ( B = c <-> u = v ) ) )
17		biimp 205	. . . . . . . . . . . 12 |- ( ( B = c <-> u = v ) -> ( B = c -> u = v ) )
18	17	necon3d 2815	. . . . . . . . . . 11 |- ( ( B = c <-> u = v ) -> ( u =/= v -> B =/= c ) )
19	18	com12 32	. . . . . . . . . 10 |- ( u =/= v -> ( ( B = c <-> u = v ) -> B =/= c ) )
20	19	3ad2ant3 1084	. . . . . . . . 9 |- ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( u e. ( EE ` N ) /\ v e. ( EE ` N ) ) /\ u =/= v ) -> ( ( B = c <-> u = v ) -> B =/= c ) )
21	20	adantr 481	. . . . . . . 8 |- ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( u e. ( EE ` N ) /\ v e. ( EE ` N ) ) /\ u =/= v ) /\ c e. ( EE ` N ) ) -> ( ( B = c <-> u = v ) -> B =/= c ) )
22	16, 21	syld 47	. . . . . . 7 |- ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( u e. ( EE ` N ) /\ v e. ( EE ` N ) ) /\ u =/= v ) /\ c e. ( EE ` N ) ) -> ( <. B , c >.Cgr <. u , v >. -> B =/= c ) )
23	22	anim2d 589	. . . . . 6 |- ( ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( u e. ( EE ` N ) /\ v e. ( EE ` N ) ) /\ u =/= v ) /\ c e. ( EE ` N ) ) -> ( ( B Btwn <. A , c >. /\ <. B , c >.Cgr <. u , v >. ) -> ( B Btwn <. A , c >. /\ B =/= c ) ) )
24	23	reximdva 3017	. . . . 5 |- ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( u e. ( EE ` N ) /\ v e. ( EE ` N ) ) /\ u =/= v ) -> ( E. c e. ( EE ` N ) ( B Btwn <. A , c >. /\ <. B , c >.Cgr <. u , v >. ) -> E. c e. ( EE ` N ) ( B Btwn <. A , c >. /\ B =/= c ) ) )
25	9, 24	mpd 15	. . . 4 |- ( ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) /\ ( u e. ( EE ` N ) /\ v e. ( EE ` N ) ) /\ u =/= v ) -> E. c e. ( EE ` N ) ( B Btwn <. A , c >. /\ B =/= c ) )
26	25	3exp 1264	. . 3 |- ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) -> ( ( u e. ( EE ` N ) /\ v e. ( EE ` N ) ) -> ( u =/= v -> E. c e. ( EE ` N ) ( B Btwn <. A , c >. /\ B =/= c ) ) ) )
27	26	rexlimdvv 3037	. 2 |- ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) -> ( E. u e. ( EE ` N ) E. v e. ( EE ` N ) u =/= v -> E. c e. ( EE ` N ) ( B Btwn <. A , c >. /\ B =/= c ) ) )
28	2, 27	mpd 15	1 |- ( ( N e. NN /\ A e. ( EE ` N ) /\ B e. ( EE ` N ) ) -> E. c e. ( EE ` N ) ( B Btwn <. A , c >. /\ B =/= c ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   <.cop 4183   class class class wbr 4653   ` cfv 5888   NNcn 11020   EEcee 25768   Btwn cbtwn 25769  Cgrccgr 25770

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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  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-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-sum 14417  df-ee 25771  df-btwn 25772  df-cgr 25773

This theorem is referenced by:  ifscgr  32151  cgrxfr  32162  btwnconn3  32210  broutsideof3  32233