Theorem aptisr 7818

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 7766	. . 3 |- R. = ( ( P. X. P. ) /. ~R )
2		breq1 4028	. . . . . 6 |- ( [ <. x , y >. ] ~R = A -> ( [ <. x , y >. ] ~R <R [ <. z , w >. ] ~R <-> A <R [ <. z , w >. ] ~R ) )
3		breq2 4029	. . . . . 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 2196	. . . 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 4029	. . . . . 6 |- ( [ <. z , w >. ] ~R = B -> ( A <R [ <. z , w >. ] ~R <-> A <R B ) )
9		breq1 4028	. . . . . 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 2199	. . . 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 7617	. . . . . . . . 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 7617	. . . . . . . . 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 4038	. . . . . . 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 7576	. . . . . . 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 7576	. . . . . . 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 7680	. . . . . . 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 7786	. . . . . 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 7786	. . . . . . 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 7775	. . . 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 6657	. 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 2160   <.cop 3617   class class class wbr 4025  (class class class)co 5904   [cec 6565   P.cnp 7330   +P. cpp 7332   <P cltp 7334   ~R cer 7335   R.cnr 7336   <R cltr 7342

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 2162  ax-14 2163  ax-ext 2171  ax-coll 4140  ax-sep 4143  ax-nul 4151  ax-pow 4199  ax-pr 4234  ax-un 4458  ax-setind 4561  ax-iinf 4612

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 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2758  df-sbc 2982  df-csb 3077  df-dif 3150  df-un 3152  df-in 3154  df-ss 3161  df-nul 3442  df-pw 3599  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3832  df-int 3867  df-iun 3910  df-br 4026  df-opab 4087  df-mpt 4088  df-tr 4124  df-eprel 4314  df-id 4318  df-po 4321  df-iso 4322  df-iord 4391  df-on 4393  df-suc 4396  df-iom 4615  df-xp 4657  df-rel 4658  df-cnv 4659  df-co 4660  df-dm 4661  df-rn 4662  df-res 4663  df-ima 4664  df-iota 5203  df-fun 5244  df-fn 5245  df-f 5246  df-f1 5247  df-fo 5248  df-f1o 5249  df-fv 5250  df-ov 5907  df-oprab 5908  df-mpo 5909  df-1st 6173  df-2nd 6174  df-recs 6338  df-irdg 6403  df-1o 6449  df-2o 6450  df-oadd 6453  df-omul 6454  df-er 6567  df-ec 6569  df-qs 6573  df-ni 7343  df-pli 7344  df-mi 7345  df-lti 7346  df-plpq 7383  df-mpq 7384  df-enq 7386  df-nqqs 7387  df-plqqs 7388  df-mqqs 7389  df-1nqqs 7390  df-rq 7391  df-ltnqqs 7392  df-enq0 7463  df-nq0 7464  df-0nq0 7465  df-plq0 7466  df-mq0 7467  df-inp 7505  df-iplp 7507  df-iltp 7509  df-enr 7765  df-nr 7766  df-ltr 7769

This theorem is referenced by:  axpre-apti  7924