Theorem aptisr 7839

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 7787	. . 3 |- R. = ( ( P. X. P. ) /. ~R )
2		breq1 4032	. . . . . 6 |- ( [ <. x , y >. ] ~R = A -> ( [ <. x , y >. ] ~R <R [ <. z , w >. ] ~R <-> A <R [ <. z , w >. ] ~R ) )
3		breq2 4033	. . . . . 6 |- ( [ <. x , y >. ] ~R = A -> ( [ <. z , w >. ] ~R <R [ <. x , y >. ] ~R <-> [ <. z , w >. ] ~R <R A ) )
4	2, 3	orbi12d 794	. . . . 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 668	. . . 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 2200	. . . 4 |- ( [ <. x , y >. ] ~R = A -> ( [ <. x , y >. ] ~R = [ <. z , w >. ] ~R <-> A = [ <. z , w >. ] ~R ) )
7	5, 6	imbi12d 234	. . 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 4033	. . . . . 6 |- ( [ <. z , w >. ] ~R = B -> ( A <R [ <. z , w >. ] ~R <-> A <R B ) )
9		breq1 4032	. . . . . 6 |- ( [ <. z , w >. ] ~R = B -> ( [ <. z , w >. ] ~R <R A <-> B <R A ) )
10	8, 9	orbi12d 794	. . . . 5 |- ( [ <. z , w >. ] ~R = B -> ( ( A <R [ <. z , w >. ] ~R \/ [ <. z , w >. ] ~R <R A ) <-> ( A <R B \/ B <R A ) ) )
11	10	notbid 668	. . . 4 |- ( [ <. z , w >. ] ~R = B -> ( -. ( A <R [ <. z , w >. ] ~R \/ [ <. z , w >. ] ~R <R A ) <-> -. ( A <R B \/ B <R A ) ) )
12		eqeq2 2203	. . . 4 |- ( [ <. z , w >. ] ~R = B -> ( A = [ <. z , w >. ] ~R <-> A = B ) )
13	11, 12	imbi12d 234	. . 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 7638	. . . . . . . . 9 |- ( ( y e. P. /\ z e. P. ) -> ( y +P. z ) = ( z +P. y ) )
15	14	ad2ant2lr 510	. . . . . . . 8 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( y +P. z ) = ( z +P. y ) )
16		addcomprg 7638	. . . . . . . . 9 |- ( ( x e. P. /\ w e. P. ) -> ( x +P. w ) = ( w +P. x ) )
17	16	ad2ant2rl 511	. . . . . . . 8 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( x +P. w ) = ( w +P. x ) )
18	15, 17	breq12d 4042	. . . . . . 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 791	. . . . . 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 668	. . . . 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 7597	. . . . . . 7 |- ( ( x e. P. /\ w e. P. ) -> ( x +P. w ) e. P. )
22	21	ad2ant2rl 511	. . . . . 6 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( x +P. w ) e. P. )
23		addclpr 7597	. . . . . . 7 |- ( ( y e. P. /\ z e. P. ) -> ( y +P. z ) e. P. )
24	23	ad2ant2lr 510	. . . . . 6 |- ( ( ( x e. P. /\ y e. P. ) /\ ( z e. P. /\ w e. P. ) ) -> ( y +P. z ) e. P. )
25		aptipr 7701	. . . . . . 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 1207	. . . . . 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 411	. . . . 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 170	. . . 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 7807	. . . . . 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 7807	. . . . . . 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 268	. . . . . 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 794	. . . . 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 668	. . . 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 7796	. . . 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 203	. . 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 6677	. 2 |- ( ( A e. R. /\ B e. R. ) -> ( -. ( A <R B \/ B <R A ) -> A = B ) )
37	36	3impia 1202	1 |- ( ( A e. R. /\ B e. R. /\ -. ( A <R B \/ B <R A ) ) -> A = B )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 709   /\ w3a 980   = wceq 1364   e. wcel 2164   <.cop 3621   class class class wbr 4029  (class class class)co 5918   [cec 6585   P.cnp 7351   +P. cpp 7353   <P cltp 7355   ~R cer 7356   R.cnr 7357   <R cltr 7363

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-eprel 4320  df-id 4324  df-po 4327  df-iso 4328  df-iord 4397  df-on 4399  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-irdg 6423  df-1o 6469  df-2o 6470  df-oadd 6473  df-omul 6474  df-er 6587  df-ec 6589  df-qs 6593  df-ni 7364  df-pli 7365  df-mi 7366  df-lti 7367  df-plpq 7404  df-mpq 7405  df-enq 7407  df-nqqs 7408  df-plqqs 7409  df-mqqs 7410  df-1nqqs 7411  df-rq 7412  df-ltnqqs 7413  df-enq0 7484  df-nq0 7485  df-0nq0 7486  df-plq0 7487  df-mq0 7488  df-inp 7526  df-iplp 7528  df-iltp 7530  df-enr 7786  df-nr 7787  df-ltr 7790

This theorem is referenced by:  axpre-apti  7945