Theorem aptisr 6863

Description: Apartness of signed reals is tight. (Contributed by Jim Kingdon, 29-Jan-2020.)

Assertion

Ref	Expression
aptisr	|- ( ( A e. R. /\ B e. R. /\ -. ( A <R B \/ B <R A ) ) -> A = B )

Proof of Theorem aptisr

Dummy variables w x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-nr 6812	. . 3 |- R. = ( ( P. X. P. ) /. ~R )
2		breq1 3767	. . . . . 6 |- ( [ <. x , y >. ] ~R = A -> ( [ <. x , y >. ] ~R <R [ <. z , w >. ] ~R <-> A <R [ <. z , w >. ] ~R ) )
3		breq2 3768	. . . . . 6 |- ( [ <. x , y >. ] ~R = A -> ( [ <. z , w >. ] ~R <R [ <. x , y >. ] ~R <-> [ <. z , w >. ] ~R <R A ) )
4	2, 3	orbi12d 707	. . . . 5 |- ( [ <. x , y >. ] ~R = A -> ( ( [ <. x , y >. ] ~R <R [ <. z , w >. ] ~R \/ [ <. z , w >. ] ~R <R [ <. x , y >. ] ~R ) <-> ( A <R [ <. z , w >. ] ~R \/ [ <. z , w >. ] ~R <R A ) ) )
5	4	notbid 592	. . . 4 |- ( [ <. x , y >. ] ~R = A -> ( -. ( [ <. x , y >. ] ~R <R [ <. z , w >. ] ~R \/ [ <. z , w >. ] ~R <R [ <. x , y >. ] ~R ) <-> -. ( A <R [ <. z , w >. ] ~R \/ [ <. z , w >. ] ~R <R A ) ) )
6		eqeq1 2046	. . . 4 |- ( [ <. x , y >. ] ~R = A -> ( [ <. x , y >. ] ~R = [ <. z , w >. ] ~R <-> A = [ <. z , w >. ] ~R ) )
7	5, 6	imbi12d 223	. . 3 |- ( [ <. x , y >. ] ~R = A -> ( ( -. ( [ <. x , y >. ] ~R <R [ <. z , w >. ] ~R \/ [ <. z , w >. ] ~R <R [ <. x , y >. ] ~R ) -> [ <. x , y >. ] ~R = [ <. z , w >. ] ~R ) <-> ( -. ( A <R [ <. z , w >. ] ~R \/ [ <. z , w >. ] ~R <R A ) -> A = [ <. z , w >. ] ~R ) ) )
8		breq2 3768	. . . . . 6 |- ( [ <. z , w >. ] ~R = B -> ( A <R [ <. z , w >. ] ~R <-> A <R B ) )
9		breq1 3767	. . . . . 6 |- ( [ <. z , w >. ] ~R = B -> ( [ <. z , w >. ] ~R <R A <-> B <R A ) )
10	8, 9	orbi12d 707	. . . . 5 |- ( [ <. z , w >. ] ~R = B -> ( ( A <R [ <. z , w >. ] ~R \/ [ <. z , w >. ] ~R <R A ) <-> ( A <R B \/ B <R A ) ) )
11	10	notbid 592	. . . 4 |- ( [ <. z , w >. ] ~R = B -> ( -. ( A <R [ <. z , w >. ] ~R \/ [ <. z , w >. ] ~R <R A ) <-> -. ( A <R B \/ B <R A ) ) )
12		eqeq2 2049	. . . 4 |- ( [ <. z , w >. ] ~R = B -> ( A = [ <. z , w >. ] ~R <-> A = B ) )
13	11, 12	imbi12d 223	. . 3 |- ( [ <. z , w >. ] ~R = B -> ( ( -. ( A <R [ <. z , w >. ] ~R \/ [ <. z , w >. ] ~R <R A ) -> A = [ <. z , w >. ] ~R ) <-> ( -. ( A <R B \/ B <R A ) -> A = B ) ) )
14		addcomprg 6676	. . . . . . . . 9 |- ( ( y e. P. /\ z e. P. ) -> ( y +P. z ) = ( z +P. y ) )
15	14	ad2ant2lr 479	. . . . . . . 8 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( y +P. z ) = ( z +P. y ) )
16		addcomprg 6676	. . . . . . . . 9 |- ( ( x e. P. /\ w e. P. ) -> ( x +P. w ) = ( w +P. x ) )
17	16	ad2ant2rl 480	. . . . . . . 8 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( x +P. w ) = ( w +P. x ) )
18	15, 17	breq12d 3777	. . . . . . 7 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( ( y +P. z ) <P ( x +P. w ) <-> ( z +P. y ) <P ( w +P. x ) ) )
19	18	orbi2d 704	. . . . . 6 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( ( ( x +P. w ) <P ( y +P. z ) \/ ( y +P. z ) <P ( x +P. w ) ) <-> ( ( x +P. w ) <P ( y +P. z ) \/ ( z +P. y ) <P ( w +P. x ) ) ) )
20	19	notbid 592	. . . . 5 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( -. ( ( x +P. w ) <P ( y +P. z ) \/ ( y +P. z ) <P ( x +P. w ) ) <-> -. ( ( x +P. w ) <P ( y +P. z ) \/ ( z +P. y ) <P ( w +P. x ) ) ) )
21		addclpr 6635	. . . . . . 7 |- ( ( x e. P. /\ w e. P. ) -> ( x +P. w ) e. P. )
22	21	ad2ant2rl 480	. . . . . 6 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( x +P. w ) e. P. )
23		addclpr 6635	. . . . . . 7 |- ( ( y e. P. /\ z e. P. ) -> ( y +P. z ) e. P. )
24	23	ad2ant2lr 479	. . . . . 6 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( y +P. z ) e. P. )
25		aptipr 6739	. . . . . . 7 |- ( ( ( x +P. w ) e. P. /\ ( y +P. z ) e. P. /\ -. ( ( x +P. w ) <P ( y +P. z ) \/ ( y +P. z ) <P ( x +P. w ) ) ) -> ( x +P. w ) = ( y +P. z ) )
26	25	3expia 1106	. . . . . 6 |- ( ( ( x +P. w ) e. P. /\ ( y +P. z ) e. P. ) -> ( -. ( ( x +P. w ) <P ( y +P. z ) \/ ( y +P. z ) <P ( x +P. w ) ) -> ( x +P. w ) = ( y +P. z ) ) )
27	22, 24, 26	syl2anc 391	. . . . 5 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( -. ( ( x +P. w ) <P ( y +P. z ) \/ ( y +P. z ) <P ( x +P. w ) ) -> ( x +P. w ) = ( y +P. z ) ) )
28	20, 27	sylbird 159	. . . 4 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( -. ( ( x +P. w ) <P ( y +P. z ) \/ ( z +P. y ) <P ( w +P. x ) ) -> ( x +P. w ) = ( y +P. z ) ) )
29		ltsrprg 6832	. . . . . 6 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( [ <. x , y >. ] ~R <R [ <. z , w >. ] ~R <-> ( x +P. w ) <P ( y +P. z ) ) )
30		ltsrprg 6832	. . . . . . 7 |- ( ( ( z e. P. /\ w e. P. ) /\ ( x e. P. /\ y e. P. ) ) -> ( [ <. z , w >. ] ~R <R [ <. x , y >. ] ~R <-> ( z +P. y ) <P ( w +P. x ) ) )
31	30	ancoms 255	. . . . . 6 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( [ <. z , w >. ] ~R <R [ <. x , y >. ] ~R <-> ( z +P. y ) <P ( w +P. x ) ) )
32	29, 31	orbi12d 707	. . . . 5 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( ( [ <. x , y >. ] ~R <R [ <. z , w >. ] ~R \/ [ <. z , w >. ] ~R <R [ <. x , y >. ] ~R ) <-> ( ( x +P. w ) <P ( y +P. z ) \/ ( z +P. y ) <P ( w +P. x ) ) ) )
33	32	notbid 592	. . . 4 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( -. ( [ <. x , y >. ] ~R <R [ <. z , w >. ] ~R \/ [ <. z , w >. ] ~R <R [ <. x , y >. ] ~R ) <-> -. ( ( x +P. w ) <P ( y +P. z ) \/ ( z +P. y ) <P ( w +P. x ) ) ) )
34		enreceq 6821	. . . 4 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( [ <. x , y >. ] ~R = [ <. z , w >. ] ~R <-> ( x +P. w ) = ( y +P. z ) ) )
35	28, 33, 34	3imtr4d 192	. . 3 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( -. ( [ <. x , y >. ] ~R <R [ <. z , w >. ] ~R \/ [ <. z , w >. ] ~R <R [ <. x , y >. ] ~R ) -> [ <. x , y >. ] ~R = [ <. z , w >. ] ~R ) )
36	1, 7, 13, 35	2ecoptocl 6194	. 2 |- ( ( A e. R. /\ B e. R. ) -> ( -. ( A <R B \/ B <R A ) -> A = B ) )
37	36	3impia 1101	1 |- ( ( A e. R. /\ B e. R. /\ -. ( A <R B \/ B <R A ) ) -> A = B )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 97   <-> wb 98   \/ wo 629   /\ w3a 885   = wceq 1243   e. wcel 1393   <.cop 3378   class class class wbr 3764  (class class class)co 5512   [cec 6104   P.cnp 6389   +P. cpp 6391   <P cltp 6393   ~R cer 6394   R.cnr 6395   <R cltr 6401

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311

This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-1o 6001  df-2o 6002  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6402  df-pli 6403  df-mi 6404  df-lti 6405  df-plpq 6442  df-mpq 6443  df-enq 6445  df-nqqs 6446  df-plqqs 6447  df-mqqs 6448  df-1nqqs 6449  df-rq 6450  df-ltnqqs 6451  df-enq0 6522  df-nq0 6523  df-0nq0 6524  df-plq0 6525  df-mq0 6526  df-inp 6564  df-iplp 6566  df-iltp 6568  df-enr 6811  df-nr 6812  df-ltr 6815

This theorem is referenced by:  axpre-apti  6959