Theorem aptisr 7720

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 7668	. . 3 |- R. = ( ( P. X. P. ) /. ~R )
2		breq1 3985	. . . . . 6 |- ( [ <. x , y >. ] ~R = A -> ( [ <. x , y >. ] ~R <R [ <. z , w >. ] ~R <-> A <R [ <. z , w >. ] ~R ) )
3		breq2 3986	. . . . . 6 |- ( [ <. x , y >. ] ~R = A -> ( [ <. z , w >. ] ~R <R [ <. x , y >. ] ~R <-> [ <. z , w >. ] ~R <R A ) )
4	2, 3	orbi12d 783	. . . . 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 657	. . . 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 2172	. . . 4 |- ( [ <. x , y >. ] ~R = A -> ( [ <. x , y >. ] ~R = [ <. z , w >. ] ~R <-> A = [ <. z , w >. ] ~R ) )
7	5, 6	imbi12d 233	. . 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 3986	. . . . . 6 |- ( [ <. z , w >. ] ~R = B -> ( A <R [ <. z , w >. ] ~R <-> A <R B ) )
9		breq1 3985	. . . . . 6 |- ( [ <. z , w >. ] ~R = B -> ( [ <. z , w >. ] ~R <R A <-> B <R A ) )
10	8, 9	orbi12d 783	. . . . 5 |- ( [ <. z , w >. ] ~R = B -> ( ( A <R [ <. z , w >. ] ~R \/ [ <. z , w >. ] ~R <R A ) <-> ( A <R B \/ B <R A ) ) )
11	10	notbid 657	. . . 4 |- ( [ <. z , w >. ] ~R = B -> ( -. ( A <R [ <. z , w >. ] ~R \/ [ <. z , w >. ] ~R <R A ) <-> -. ( A <R B \/ B <R A ) ) )
12		eqeq2 2175	. . . 4 |- ( [ <. z , w >. ] ~R = B -> ( A = [ <. z , w >. ] ~R <-> A = B ) )
13	11, 12	imbi12d 233	. . 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 7519	. . . . . . . . 9 |- ( ( y e. P. /\ z e. P. ) -> ( y +P. z ) = ( z +P. y ) )
15	14	ad2ant2lr 502	. . . . . . . 8 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( y +P. z ) = ( z +P. y ) )
16		addcomprg 7519	. . . . . . . . 9 |- ( ( x e. P. /\ w e. P. ) -> ( x +P. w ) = ( w +P. x ) )
17	16	ad2ant2rl 503	. . . . . . . 8 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( x +P. w ) = ( w +P. x ) )
18	15, 17	breq12d 3995	. . . . . . 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 780	. . . . . 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 657	. . . . 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 7478	. . . . . . 7 |- ( ( x e. P. /\ w e. P. ) -> ( x +P. w ) e. P. )
22	21	ad2ant2rl 503	. . . . . 6 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( x +P. w ) e. P. )
23		addclpr 7478	. . . . . . 7 |- ( ( y e. P. /\ z e. P. ) -> ( y +P. z ) e. P. )
24	23	ad2ant2lr 502	. . . . . 6 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( y +P. z ) e. P. )
25		aptipr 7582	. . . . . . 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 1195	. . . . . 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 409	. . . . 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 169	. . . 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 7688	. . . . . 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 7688	. . . . . . 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 266	. . . . . 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 783	. . . . 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 657	. . . 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 7677	. . . 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 202	. . 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 6589	. 2 |- ( ( A e. R. /\ B e. R. ) -> ( -. ( A <R B \/ B <R A ) -> A = B ) )
37	36	3impia 1190	1 |- ( ( A e. R. /\ B e. R. /\ -. ( A <R B \/ B <R A ) ) -> A = B )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 698   /\ w3a 968   = wceq 1343   e. wcel 2136   <.cop 3579   class class class wbr 3982  (class class class)co 5842   [cec 6499   P.cnp 7232   +P. cpp 7234   <P cltp 7236   ~R cer 7237   R.cnr 7238   <R cltr 7244

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-eprel 4267  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-irdg 6338  df-1o 6384  df-2o 6385  df-oadd 6388  df-omul 6389  df-er 6501  df-ec 6503  df-qs 6507  df-ni 7245  df-pli 7246  df-mi 7247  df-lti 7248  df-plpq 7285  df-mpq 7286  df-enq 7288  df-nqqs 7289  df-plqqs 7290  df-mqqs 7291  df-1nqqs 7292  df-rq 7293  df-ltnqqs 7294  df-enq0 7365  df-nq0 7366  df-0nq0 7367  df-plq0 7368  df-mq0 7369  df-inp 7407  df-iplp 7409  df-iltp 7411  df-enr 7667  df-nr 7668  df-ltr 7671

This theorem is referenced by:  axpre-apti  7826